Not sure there is much in the way of airport hotel options at the airport - I haven't tried, but can't imagine there's one right at the airport - all probably involve a bus anyway and the bus to Basel runs until midnight so I'd think you're better off getting that and being downtown then.
Any time I've flown there arriving late (1030pm or so from London) it's quite busy in the airport .. It's a fairly small airport, but not tiny - The airport used to be segregated into a Swiss side and a French Side and you couldn't cross between them, but this is no longer the case (there's still signage indicating it, but once landside you can freely walk from one side to the other.)