The official name of the event is Bay to Breakers, as the race/event begins on the shore of San Francisco Bay and runs west 7.5 miles to end at the Pacific Ocean (the "breakers"). There really IS a race, though the race usually has ended just as the "party" is getting under way. I've lived in the San Francisco area since 1967, and would have to venture a guess that the party aspect of B2B started in the late '70s. But then, marathon races hold as much interest for me as watching paint dry.
There is a group that sponsors and organizes the "Bare to Breakers" naked people, with its own website,
http://baretobreakers.com/. If you sign up with them and pay their fee, you get a yellowish-green (greenish yellow?) baseball-style cap with the Bare to Breakers logo to wear. They say they've been around since 1993, so I guess that's when the nudity began. They also admit female participation is only 20%.