This is a very good question, and it remains a very good question to ask despite the fact that Popkin's question has pretty much been answered several times over. This is one of the things that historians of philosophy do: we establish topographies of evidences. Even if you know that Hume had reading knowledge of so-and-so, it's still worthwhile to know exactly what supports the claim that he did so. One reason, obviously, is that you can't actually know that Hume had reading knowledge of so-and-so without knowing the evidence for it; but another reason is that the evidences sometimes highlight features of Hume's work that might go unrecognized if you don't realize that the evidences are there in the first place. This is precisely what happened: Hume scholars went to work answering Popkin's question. Actually, almost no one thought that Popkin was right in suggesting Hume's ignorance, but forcing scholars to lay out in a clear and articulated way why he was wrong led to all sorts of discoveries about Hume's work.
There are a number of internal evidences for Hume's real acquaintance with Berkeley. For instance, Hume footnotes Berkeley in a passage in the Treatise (1.1.7, on abstract ideas) that has close connections to the Principles. You might not put much emphasis on the footnotes, although Hume is so stingy when it comes to acknowledging influences (this is not unusual in the period) that one could reasonably argue that a footnote acknowledging the importance of someone is more than just a casual mention in passing. But one definitely does want more, and there are internal evidences aplenty. In the essay "Of National Characters" he paraphrases a passage from Berkeley's Alciphron (and attributes the idea to Berkeley in a footnote); the passage is buried deep in Dialogue V of Berkeley's work, so that suggests it was not from a light reading. More than this, however, it is possible to find what are likely echoes of Berkeley in Hume's discussions of our knowledge of bodies, sensible minima, the distinction between primary and secondary qualities, his account of mind, and others. Careful sifting by scholars still turns up new plausible cases even today.
The internal evidence, then, is quite good; much better than Popkin had suggested, although Popkin was quite right that most of it is not on the immediate surface. To find it you have to identify echoes of phrasing, parallel structures of argument, and ideas original to Berkeley that are also found in Hume. This takes quite a bit of comparative work to do properly, and, indeed, much of it has taken decades of serious work and considerable debate. But an additional problem with internal evidences in general is that they admit of alternative explanations. Berkeley's main works were talked about quite a bit; it's always possible that genuinely Berkeleyan ideas were the topics of conversation and thence made it into Hume. In this sort of case, Hume would have known genuinely Berkeleyan arguments and ideas, but the kind of transmission would be different. This would be important to know, because secondhand oral transmission of philosophical ideas and arguments works rather differently than transmission by direct access to philosophical texts. Other explanations also can arise: sometimes ideas are 'in the air', i.e., due to common environmental or social causes; sometimes common ideas in two authors indicate not influence between them but influence from a common source (who may or may not be already known); and so forth. Given the extent of the internal evidence that Hume was acquainted with details of Berkeley's work, the scale on which one would have to deploy these alternative explanations would be extraordinary and implausible. But the internal evidences themselves don't fully rule them out.
What one needs is a direct link, and in the Berkeley-Hume case, the first real direct link was the purported discovery of a letter from Hume to Michael Ramsay dated August 31, 1737; for purposes of Hume scholarship, arguably the most important letter from Hume ever discovered. In this letter, Hume, who is about to publish the first book of the Treatise of Human Nature recommends some background reading for Michael Ramsay to understand the metaphysics of the work. He recommends, specifically, Malebranche's Search after Truth, Berkeley's Principles of Human Knowledge, the more metaphysical articles in Bayle's Dictionary, such as the articles on Zeno and Spinoza, and, if Ramsay could at all find it, Descartes's Meditations on First Philosophy. Since Hume is quite clear that he was directly influenced by these works, it is, as one might say, a 'smoking gun'. The Ramsay letter is one of those far-flung lines of evidence that are often found in the history of philosophy. A Polish princess, Princess Izabella, had acquired it. She was actually Scottish herself, Isabella Fleming by birth, but had married to Prince Adam Kazimierz Czartoryska. At some point the princess had met David Hume the younger -- the nephew of David Hume the philosopher and historian -- and she made a serious effort to obtain manuscripts of the original Hume. In 1790 she acquired five letters of Hume, one of which was the Ramsay letter. Thus the direct evidence for whether the Scottish philosopher had ever closely read the Irish philosopher was hanging out in the Czartoryski Museum in Krakow, Poland. Tadeusz Kozanecki published three of the letters in a Polish journal in 1963; Popkin became aware of this and in 1964 conceded the point in the article, "So, Hume Did Read Berkeley".
In 1973, Michael Morrisroe published a potentially even better 'smoking gun' evidence. In another letter to Michael Ramsay, this one dated September 29, 1734, Hume explicitly says that he was re-reading Berkeley's Principles and Locke's Essay. You can't get plainer than that. Unfortunately, things get a little complicated. Morrisroe tells us that he was given the opportunity to make a typescript, but that the letter was auctioned off and its location unknown. He did not say what he did to establish authenticity, although the letter (about ten sentences long) sounds broadly Hume-like. As far as I know Morrisroe's source has never been rediscovered. It was generally accepted as legitimate, but the circumstances put it in a very different category of evidence than the 1737 Ramsay letter. Hume scholars have generally been cautious with it; it primarily only gets mentioned in biographical contexts, and in recent years there has been some more skepticism about it. This letter from Morrisroe is the letter that Waldmann has argued, based on its mention of Pluche, cannot be authentic.
In 2011, while preparing a 1709 edition (which was he second edition) of Berkeley's An Essay Toward a New Theory of Vision for an exhibit, Donald Kerr at the University of Otago discovered that the book had a David Hume bookplate. Peter Anstey discussed it online not long after. This gives us a different kind of evidence from a letter. We have here a book with a bookplate. The bookplate establishes at the very least that the book was in the library of one of the two David Humes (the philosopher or his nephew); and the bookplate in question is one of two different David Hume bookplates. If we add to this the Hillyard-Norton hypothesis that the State A bookplate is that of the original David Hume, we get the result that Hume had this work in his library, and thus had at least the opportunity to have read it; and given Hume's penchant for reading, it increases the likelihood that he read this particular work. This of course has a certain tenuousness to it. There are good reasons to accept the Hillyard-Nortan hypothesis, but they are all indirect; and merely having the book on the shelf isn't an automatic guarantee of having read it. (I mean, I am a voracious reader, usually reading several books a week, and I've had books on my shelf for ten years that I have still not gotten around to reading.) But, as I noted before, one of the topics on which internal evidence suggests that Hume was influenced by Berkeley is the topic of sensible minima, so this discovery immediately suggests the project of looking more closely for parallels, echoes, and the like connecting this particular work with Hume's discussion of that topic.
Of course, when we are talking about whether Hume read Berkeley, we really mean several different works by Berkeley, and this needs to be taken into account. The result we actually have is, roughly:
(1) Hume certainly read the Principles, and very early on; the internal evidence supports this very strongly already, and if even one of the Ramsay letters mentioned above is genuine, the evidence establishes this as much as anything of this sort can be established.
(2) Hume may have read the Three Dialogues at some point, but this is only suggested by occasional internal evidence -- passages in Hume's works that are reminiscent of things in the Three Dialogues.
(3) Hume very likely read the Alciphron, on the basis of the fact that he paraphrases and refers to a particular passage in it in the essay "Of National Characters"; this, the strongest of the internal evidences, would put the reading of the Alciphron before 1752, when the essay began to be published in Hume's essay collections. Unfortunately, as the passage in question is a pretty trivial passage and is incidental to Berkeley's overall argument, it doesn't tell us much at all about how closely Hume read this dialogue, beyond the fact that he seems at least to have been struck by a figure of speech in Dialogue V. It does, however, establish that he cracked the book and read some of it. Possible lines of further study are links between the Alciphron and Hume's economic essays and links between it and the Dialogues Concerning Natural Religion.
(4) Hume could well have read the New Theory of Vision, since he likely had it in his library, due to the bookplate evidence; some internal evidence is at least consistent with this, and a closer look at internal evidences relevant to the topics in this work is thus warranted.
(5) People have argued that there is some internal evidence of the influence of Berkeley's Querist on Hume's economic essays, although I don't know how strong some of the suggested evidence actually is.
(6) Of a number of Berkeley's other works -- Siris, the Theory of Vision Vindicated, etc. -- there is currently no evidence (so far as I know) of influence, although , of course, new evidence could always turn up. Hume was at least aware of the basic idea of Passive Obedience, i.e., passive obedience, since he mentions it, but I don't know of any work done on direct links or influences, and this is very much one of the ideas that would have been talked about anyway.
All this is somewhat simplified. Getting this far really involves a great deal of argument back and forth among scholars as they try out the various ramifications of looking at the evidence this way and that. But it serves to give an idea of how the finding, sorting, filtering, and integrating of evidence works in historical approaches to philosophy.