The only problem that I have (beyond the basic "EWWWW" factor) is the occasional degredation of sound from playing too long without clearing.

Worst case scenario, buy some pipe cleaners. If you feel that the whistle is becoming clogged, you can quickly clean the fipple out with the pipe cleaner between songs.

Further, I do know a couple players who when they're sitting during a gig will drape a towel across their leg at an appropriate point.

BTW - this is a problem with pretty much ALL wind instruments or brass instruments I've heard of. Most of the more complex will have a 'spit valve' for drainage. The simpler just let it fall where it may... my mom has wonderful stories of her band days with the joys of flutes...

M