[perhaps you're looking for a boat that actually moves, but] the two or three boats on the North bank of the Thames between B'friars bridge and Hungerford Bridge always used to be open to anyone (I haven't been on any of them for a few years).
If you find a moving, floating, pub-boat, then do let us all know please.
There is a floating pub down at Greenland Dock (it used to be a ferry somewhere but it doesn't go anywhere now). It's called the Wibbly Wobbly (!) and seems to have normal pub opening times; also serves food. It's at the far end, near to the lock gates on the South side.
The ones on the north bank are still open to the public - there is the Queen Mary by Waterloo Bridge, the Hispaniola by the Hungerford Bridge and the Tattershall Castle, plus another boat further toward Blackfriars Bridge which I forget the name of, but it does Thai food and I think has a bar as well as the restaurant.
There is another boat called Regalia on the North side of London Bridge. They do barbecues. You don't need to drink much, you feel very tipsy with the rocky movements of the Thames tidal waves. I guess hiring a whole boat for yourself would be very expensive and would require of minimum spent amount, especially in the summer when they host events like companies summer parties and weddings that bring in a lot of money..