And what if we think others are holding onto (say) religious or political beliefs that are dubious? Well, again the same applies: a stringent dose of scepticism is what is needed. (Which isn't to say that it is always appropriate or polite to try to administer it!) For if a belief doesn't stand up to vigorous rational scrutiny it really isn't worth having. So in fact the way to show respect for those we disagree with is indeed precisely to engage critically with their ideas and argue: which is, of course, how we learn to improve our own views. On the other hand -- and I take it that this is perhaps the thought behind the question -- the opinions of believers who try to insulate certain of their beliefs from such vigorous scrutiny, or who try to block open critical enquiry of their ideas, deserve no respect at all.
Whether this even makes sense depends very much on what is meant by "vigorous rational scrutiny" and "belief". If the first means "scrutiny showing that it is not arbitrary," which is not what we would usually mean by the phrase, and the second means "any sort of cognitive judgment in which something is definitely accepted," then it probably stands; a belief that doesn't stand up to vigorous rational scrutiny isn't worth having. Or to be more accurate, a belief that was not capable of standing up to scrutiny of this sort, whether the scrutiny was actually performed or not -- it would be absurd to demand actual scrutiny of every belief to see if it has such reasons, because then you would be demanding the impossible. You believe that your socks came from your sock drawer? What are your reasons for it? So you believe those are good reasons for it? What are your reasons for believing that those are good reasons for it? O, really; what are your reasons for believing that those reasons are good reasons for believing that the things you said were good reasons for believing that your socks came from your sock drawer really are? &c., &c.
So if we take "if a belief doesn't stand up to vigorous rational scrutiny it really isn't worth having" to mean something so weak and general, it's probably true. But we have lots of beliefs worth having that couldn't stand up to vigorous rational scrutiny if "vigorous rational scrutiny" meant something like a full-scale investigation involving "a stringent dose of scepticism", not because the beliefs themselves are bad, but because we would simply lack the resources for defending them against a "stringent dose of scepticism". How in the world could you ordinarily defend your belief that your socks came from your sock drawer from someone who was not willing to take anyone's word for it? That level of scrutiny for that sort of belief is itself absurd and irrational.
This sort of claim sounds good enough, and is almost certainly true in some meaning or other. But it depends very much on which meaning we take it to have. Likewise, the claim that "the way to show respect for those we disagree with is indeed precisely to engage critically with their ideas and argue" depends very much on the context and what is supposed to be involved in 'engaging critically with their ideas and arguing'. Someone who feels that he has to argue with everything he disagrees with until others agree with him is not respecting anyone; he is just acting like an idiot, and showing that he is more interested in his own beliefs than in the people around him. Sometimes the most respectful way to argue is simply to give your reasons for thinking otherwise, and leave it at that. Sometimes it's not reasonable or respectful to pursue an argument at all. The belief that a man was a decent and upright citizen may not withstand vigorous rational scrutiny; trying to argue his widow out of it at his funeral is hardly reasonable or respectful for all that. What counts as respect varies according to circumstances; this is true of any sort of respect. Likewise any sort of argument can be couched in insinuating or insulting terms that are not respectful, and even morally despicable, so how you go about arguing is important as well.
But in any case respect for those with whom one disagrees isn't precisely the same as respect for someone's beliefs, although the former may under certain conditions contribute to the latter. There are lots of reasons you might respect someone's belief even though you think it's wrong -- it might be very clever, very beautiful, very intelligent, very intriguing; even if probably wrong it still might in some way be worthy of serious and openminded inquiry; they might be an expert in the field, or very rational and reasonable people, and thus you would see their wrong opinion as nonetheless being one that has to be taken seriously because it's the sort of opinion rational and reasonable people are likely to have; etc. Indeed, you may respect someone's belief even when you don't respect them at all; we do it quite often, because the people we have difficulty respecting will certainly believe some things in common with ourselves. Most of us, I take it, regard our own beliefs as respectable; at least, the only time people ask whether they should respect someone's beliefs is when that someone disagrees with them.
But the question was a tricky one, particularly given how it was stated; I'm not sure precisely how I would have answered it myself. Smith was right to point out that it doesn't make much sense to see oneself as trying to "root out beliefs as much as possible". I would probably also have pointed out that, whatever may be the case for occasional contempt, regularly feeling contempt for people is neither reasonable nor morally healthy, regardless of what occasions it. And I would have pointed out that respecting someone and feeling respect for them are not the same thing; very often when you respect someone you aren't feeling anything distinctive at all, and sometimes you most definitely show respect for someone by doing so when they make it difficult to feel respect for them. And, of course, that respecting people and respecting beliefs are different things, however related they may be. It's difficult to find any fully adequate account of respect itself, however; and without that, we are still in dialectical mode whenever we talk about respecting beliefs.