The Topic was founded in September 1956, and by the miracle of statistics I see it has been open on average more than 51 weeks of the year ever since (from looking at the gig list 1970 to now and incorporating info from the Silver Jubilee history in 1981, both on the club website, it has only been closed on 33 potential club nights in 50 years, usually Xmas or New Year - it was last closed Xmas Day 2003).
Looking at the information here the San Francisco club is fairly informal ("It is not a club in the organizational sense since nobody has ever been elected to anything and there aren't any bi-laws or written rules" though despite that they do seem to organise a lot of stuff...), it is bi-weekly and in a private house. And although it was founded in 1948 perhaps - I don't know - the SFFMC's existence was patchy in the early days as it seems to have been reorganised in 1959 to ensure continuity.
There are a couple of other early claimants. The Cornell Folk Song Club in New York was founded at some point in the 50s, perhaps only as some sort of informal university society to start with (was it open to the public?), became formalised in 1957 according to its website and even now seems only to be fully-functioning during term-time. The Bush Music Club, Inc in Sydney was founded in 1954, but seems to be less a normal folk club as we'd understand it in the UK than an EFDSS-style organisation with publications along with dance and music workshops, concentrating on bush music. Links to both places on the Topic site via the History page.
So the Topic's claim to being the oldest folk club in the world - which has been on its T-shirts and publicity since September 1995 - may require certain qualifications, such as being weekly, continuously-operating, having an AGM and paid membership, getting paid guests, being in a public venue and so on. It can't claim to have been continuously in the same place since foundation as it's now on its 11th venue, nor can it claim to have been overseen by the same people from the start.
Still - the Topic is 50 years old this year, so it pre-dates the satellite era, and that's a cause for celebration and some back-slapping and thanks to now-retired veterans like Trevor Charnock and Philomena Hingston, who kept the club going in good times and not-so-good, and Roger Sutcliffe, one-time club secretary who has been performing at the club annually since 1964.
Currently times are pretty good, by the way. There'll be no specific big birthday weekend bash for the golden jubilee as there was for the silver but the birthday will be noticed at several points... and guests in autumn include Vin Garbutt, Wizz Jones, Julie Felix, Dana and Susan Robinson, Phil Hare, and Steve Gillette and Cindy Mangsen.
Though I don't get to go often as I live 00's of miles away I can't claim to be disinterested and unbiased as I do the website for the Topic for my brother John, who's act booker and occasional MC and floorspot performer there.