I'm piecing this together from a bunch of different stuff I've read and listened to, but I'd say the top contender is probably Minsky's at the Winter Palace, which started in NYC in the 1910s and 20s. Dedicated strip clubs basically evolved from burlesque.

Burlesque itself started off meaning a play or poem in which grand, epic style was applied to trivial, silly subjects --- chorus girls declaiming mock-Shakespeare or comedians dressed up like Roman Senators and making cracks about contemporary politicians. Kind of the same function an SNL skit serves today. Burlesques started out as a quick break in the intermission between acts in a serious play or variety act, but then this one London promoter got the idea of starting a burlesque troupe composed entirely of hot chicks and the thing took off like a rocket ---toured the US, the works. That started off burlesques being popular in their own right, and by the late 19th century you could see a whole show that was centered around burlesques, with maybe some comedians for a warm up act and some dancers as well. By the 1890s-early 20th century there was a fashion for "exotic dancers" --- by which was meant woman doing dances from the "exotic east" in "native costumes" --- belly dancing, fan dancing, Bollywood type stuff. (Or at least dances people in Western Europe thought seemed exotic looking.) In a time when typical women's clothing hid everything from the ankle to the neck, a belly dancing outfit was pretty scandalous, and the dancers started pushing the envelope pretty early. In Paris, you had people like Mata Hari and caberet places like the Moulin Rouge and the Follies Bergere where the shows could get pretty risqué --- topless girls in tutus, etc. In the states the burlesque shows --- which were already basically hot girls+skimpy clothes+dirty jokes, for an audience of drunk dudes---started to add in the exotic dancing, and pretty soon full on stripping was part of the show. Minsky's in NYC started out as a burlesque house --- their original theater was kind of a shit hole, so they were willing to put on more risqué acts to attract an audience. They were the first in the US to incorporate a long platform jutting out into the audience as part of the show, the same kind the Follies used in Paris. In the 1920s when prohibition hit Minsky's became a lot more popular --- they got raided so much by the cops for violating obscenity laws it was a kind of standing joke. (And the title of a movie --- The Night They Raided Minsky's.)They got shut down for good in the 30s but by then the innovations and envelope pushing they'd done were pretty standard for burlesque theatres, and you can still see the echoes in strip clubs today --- long strip of platform that the girls perform on with seats around it, sleazy MC/comedian (today, a DJ) to warm up the crowd between acts.