Plato had a famously dim view of democracy. He regarded politics as a craft, and thought that understanding the essence of a craft is to have expertise. Plato argues that we cannot hope the multitude to achieve expertise in the craft of governing. They are too easily misled by sophists. It followed, for Plato, that democracy must be rejected as a just system of governance. It is “probable that the origins of tyranny are found nowhere else than in the democratic regime.” (“The Republic”). A just system of government must have a philosopher king, who understands the essences of things. Translated into the modern context, Plato’s view is that the only just system of government is one that is run by one or several experts in economics and public policy. The multitude is too easily swayed by propaganda.
Plato was right to regard his views as inconsistent with democracy. His view that citizens are not competent to make judgments about public policy, that economics and policy are areas of expertise like the physician’s, is profoundly undemocratic.
It is true that Plato has a dim view of democracy, but this definitely involves taking Plato's comments on democracy and exaggerating them beyond all recognition. (I don't think Stanely is alone in doing this.) So some points to the contrary, with minor comment:
(1) The claim that Plato thinks "that citizens are not competent to make judgments about public policy" is the exact opposite of the view that Socrates actually presents. The structure of the kallipolis requires that citizens make lots of judgments about public policy -- namely, in those areas of public policy directly relevant to the skill by which they are able to benefit the city. Some citizens, of course, have more extensive 'jurisdiction' as we might call it, because they develop more comprehensive skills; but this is not surprising, since the structure here -- that some people have more responsibility and power to govern than others -- is true of every society with a government, and the kallipolis or city of virtue by its very nature has to be a meritocracy. Given that Socrates is answering the question of what kind of luxurious city (more on that in a moment) would be incapable of injustice, the kallipolis is a city in which all policy is public policy, because everything has to be for the good of the city. The differences are just in more-and-less: some people are capable of actions and decisions that are more beneficial for the city than others.
Likewise, it is incorrect to say that Plato's view is "that we cannot hope the multitude to achieve expertise in the craft of governing". The Platonic Socrates does indeed think that people are too easily misled by sophists, but this is not because they are incapable of the craft of governing but because they listen to flattery and pandering rather than examining their lives and trying to be just. Likewise he thinks they are enamored with democracy because of a complacent confusion of the gaudiness of mere tolerant diversity with the beauty of justice, not because they are incapable of being goaded into recognizing the difference. Since Socrates in several dialogues (the Republic is not an exception) closely links, and sometimes seems to identify, the craft of governing with the craft of self-governing, government ends up being the ability to develop justice in oneself and encourage it in others. The whole Socratic approach, however, assumes that everyone is at least in principle capable of this, and in the Republic Socrates is quite clear that the way to prevent the degeneration of society is for people to try to be the kind of citizen appropriate to a kallipolis, regardless of what their actual society might be.
The only way in which we can make these opinions work is by glossing "multitude" as "people who are not trying to know what justice is and live their lives in accordance with it"; in which case it is certainly true that Plato would think that we cannot hope for such people to achieve the skill of governing, and that they are not competent to make judgments about public policy. If we take it in that sense, though, I think we have to admit that there is something to be said in its favor.
(2) It's easy to overlook elements of the the structure of the Republic, since there is so much to the dialogue, but the most fatal element to overlook is the fact that the kallipolis is not Socrates's first choice for a just society. It is his second, the one that takes most of the book even to describe. The first just society presented in the dialogue doesn't take long to describe at all. In this first society people come together and divide the labor required to survive. For instance, instead of one person trying to do and get everything he needs, he focuses on making food for everybody and gets things like clothing by exchanging with other people. Everybody works hard and budgets thriftily and trades fairly because they have to. For social fun they eat good food in moderation, drink in moderation, and sing hymns together; they also "roast myrtle and acorn by the fire", which is sometimes taken to be a sexual euphemism. And that's it. The society is organized not by a government but by the three principles of free market, necessary thrift, and mutual aid. (Maybe you could add the principle of church picnics.) There are no guardians, no philosopher-kings. There isn't even any government in our usual sense; everybody does what they have to and injustice is avoided because it is obviously contrary to everyone's interest. Kropotkin would love it; it is a self-organizing anarchy in which injustice is impossible precisely because everyone has to govern themselves and their own families, and people work together and get along because they have to do so.
The Athenians, of course, are utterly shocked at this picture of a just society, calling it a "city of pigs", and demand that Socrates give them a just society in which the people have luxuries: quite literally, a city in which people can have comfy cushions, tables, fancy sauces, perfumes, pastries, and courtesans. In other words, they want not a just trading society, they want a just trading power. They want a city in which they can live exactly like they do in Athens but still guarantee that there will be no injustice. But how do you avoid injustice in a society in which people devote themselves to obtaining luxuries? You have to make it possible for them to have luxuries while making it at least extremely difficult for them to fall into the trap of always craving more (pleonexia, as the Greeks called it). Craving more and more is a major, perhaps the major, cause of injustice. And that is where we get the republic, the kallipolis, and that is why the kallipolis is so weird: this is how far Plato thinks you'd have to go to have city that both pursued wealth and luxury like Athens and was immune to injustices of ambition and craving. People would have to treat each other more like family than like strangers, rulers would have to regard things like bribery and sale of offices in the way they regard drinking poison, education would have to be carefully structured to weed out even the idea that people might be better off if they were unjust, etc., etc. This is also why, having built the kallipolis in idea, Socrates then goes on to show how it would slowly collapse in real life as, just by sheer accumulation of accidents, people start realizing that you can do what you're supposed to and still not get the nice things you're supposed to get for it -- and therefore start taking shortcuts.
(3) Plato does think that democracy tends naturally to tyranny, but this is closely tied to what he means by democracy. Plato criticizes democracy not because he thinks it is bad government but because he thinks it is a near-abdication of government: the principle of a democratic society is one in which everyone gets to do whatever they like, regardless of whether it benefits themselves or others, as long as they don't harm anyone else. This principle of tolerance gives order to the society, but it is a necessarily unstable order. For it to work everyone has to agree about what is harmful and what is harmless. But there is nothing in democratic society that can possibly guarantee this agreement. Even if they start out in agreement, it would break down over time as people naturally diverge and have different experiences. In order to handle this it becomes absolutely necessary to persuade other people, not about minor things, but about the most fundamental things. Thus we have the rise of sophists and orators. Democracy becomes a power struggle between interest groups, each trying to persuade a sufficient majority that they have the right idea about what is harmful and what is harmless while their opponents are really doing what is harmful. Opinions are deemed bad not on the basis of the understanding that they summarize but simply because you can find a way to bully or shame enough people into not holding them. Only the bare principle of tolerance and the balance of power prevent the society one in which might makes right.
But the democratic city also has nothing that can guarantee that balances of power will last forever or that people will continue to respect the principle of tolerance. Sooner or later the balance of power tips heavily in one direction, sooner or later one group gets enough power that they begin to realize that they don't actually need to tolerate people who disagree with them, and that life would actually be easier for them if they didn't. So they stop doing it. And then you have tyranny: a society whose only principle of rule is someone being able to manipulate others into destroying those who disagree with them.
To put it in other words: Plato does not take a dim view of democracy because he thinks ordinary people are incapable of developing the skill of governance; he takes a dim view of democracy because he thinks democracy by its nature makes it almost impossible for them to develop it. He doesn't think that democracy tends toward tyranny because people are in themselves too easily misled by sophists and orators; he thinks that it does so because democracy by its very structure makes it difficult for them not to be misled by sophists and orators. People are protected from being misled by sophists and orators in a kallipolis, because everyone has to develop at least some public-minded virtues; they are protected in a timarchy because everyone has to conform to standards of honor that are not easily twisted around; they are protected in an oligarchy because orators have to show that their proposals really do stand a chance of being useful and profitable; but they have no protection at all in what Plato calls a democracy. Plato is not skeptical of democracy because he thinks most people are incompetent when it comes to political decisions; he is skeptical of democracy because he thinks it makes most people incompetent.
(4) Nonetheless, it is always worth keeping in mind that, for all this, Plato is quite clear that democracy is still a genuine government and a society that still has some echo or shadow of justice. It is the weakest society, yes, but it is a true and genuine society, one in which we can still say there is a public good, a common good. It is salvageable (by philosophy and the pursuit of virtue) and beneficial to its members. A democractic society in Plato's sense can be (and should be) much better than it is, but the citizens still can make it better without superhuman acts of heroism. The degeneration of democracy into mobocracy and tyranny is an extraordinary catastrophe, one of the worst catastrophes possible, because it is the last government in which the society still has anything to do with justice at all.
