James Romm has a very good discussion of the Platonic Letters, What the controversial letters of Plato reveal about us, at "Aeon.co":
I cannot produce an answer to the question of whether Plato wrote some of the Platonic letters (he certainly did not write them all), nor can anyone, for no such answer is possible ‘beyond a reasonable doubt’. I rather aim to explore the history of the debate, as confidence in the letters’ authenticity has waxed and waned over centuries, reaching its lowest ebb ever in recent decades but now, perhaps, rising once again. This history reveals a pervasive bias against Platonic authorship of the letters, based on a desire – unconscious, no doubt – to distance the exalted figure of Plato from their less-than-exalted content. Several of the letters relate to an episode in Plato’s life that Plato’s admirers find troubling: a failed attempt to collaborate with Dionysius II, the immensely powerful ruler of Syracuse, in an effort to reform the government of that Sicilian Greek city.
When I did my read-through of the entire Platonic corpus nine (!) years ago, I covered the Letters, of course. I'm not competent to assess the subtler textual-critical arguments about whether (for instance) the language and style are Platonic, but I came to the conclusion that with regard to the Seventh Letter (which is the one most likely to be authentic, and which I discussed here) the only serious content-based argument against its authenticity is that its theory of Forms is slightly different from what one would expect, given Aristotle's account of Plato, although perhaps not too far removed from the kinds of examples that we actually see Plato use in the dialogues (like the notorious metaphor of the three beds in the Republic). My own assessment was analogous to Romm's here, that the primary factor in whether any given Plato scholar regarded the Seventh Letter as authentic was simply whether they wanted it to be authentic.
In any case, as I said, the textual matters I am not competent to judge, but Burnyeat's claim, mentioned by Romm, that the author of the Seventh Letter was 'philosophically incompetent' is just such utterly, blatantly nonsense that it makes me skeptical of everything Burnyeat says about the matter. Whether by Plato or just an astute Platonist, the Seventh Letter is a brilliant discussion of education and rule of law on Platonic principles, and anyone can read it with benefit. You can read it in Bury's translation at The Perseus Project.
