I've typed up a very long and meticulously cited FAQ in response to this very old charge, available here.
I think I've now read every blog post there is on the subject, and hopefully I covered just about everything. I would appreciate comments, criticisms, etc.
Basically, what happened was that for perhaps 3 decades, several newsletters ran under Paul's name containing some fairly conventional right-wing commentary. Many have attested that Paul personally distanced himself from production and various people with gigs as ghostwriters came and went.
For a brief period from about 1990-1994, a very small number of newsletters were released sporadically which contained racial hateful and "homophobic" remarks.
This was during a time when Paul had relinquished responsibility for the newsletters' operation, retired from Congress and an exhausting presidential campaign, and consigned himself to working full-time as a medical doctor and public speaker, in addition to raising five children.
A few objectionable issues managed to leak out under Paul's nose, quite understandably. Paul did not then and does not now possess superhuman powers. Believe it or not, there were other things that were occupying him at the time. He didn't have the ease of mind, the way some people apparently do, to devote his days to scanning the newsletters for the occasional rant against gays. Blaming him for this is ultimately like blaming him for comments on his Facebook wall.
The racist comments fly in the face of everything Paul has ever written and said, as many people who know him personally have attested. (Among whom include Rick Sincere, an openly gay libertarian whose run for Congress in 1993 Paul supported and helped solicit funds for, at the same time the most hysterical of the newsletters were being churned out.)
Paul was quite angry when he learned of the whole thing. He didn't issue a full denial in 1996 when it was first brought up only on the extremely stupid advice of his campaign staff. (Paul is far too trusting and has never been great at picking advisors.) He has in the past decade addressed the issue several times publically, explicitly denying authorship, and there is every reason to believe it and move on.