A couple of weeks ago in the New Yorker, Sasha Frere-Jones had an admirable go at parsing the "Complete Basement Tapes" (possible alternative title to the article: "The Basement Tapes: Their Part in My Downfall"). You can read that article here.
This is not the first time Dylan has appeared in the magazine. Ellen Willis wrote at least a couple of pieces during her stint as its rock music critic. And Nat Hentoff wrote one of the magazine's "Profiles" on him in 1964. Hentoff followed his subject around during the recording of "Another Side of Bob Dylan", although the article itself seems not to have appeared until a month or so after the album was released (if Wikipedia can be trusted). Well, things moved more slowly back then. Presumably Dylan, as interview subject, did his usual amount of dissembling: it would be interesting to know what the fabled fact-checking department made of it all.
The Hentoff piece was reprinted in the book of interviews "Dylan on Dylan", but my preference (pathetic as it may be) is to read New Yorker articles in situ, typed using the actual New Yorker typeface, and in their original context, i.e., surrounded by magazine ads of the period. (Like, uh, "Go ahead, be a narcissist". Woah.) Anyway, I made a PDF of it, and you can get that from the Dropbox, at least for a little while.