I ALWAYS have a small (dish? hand?) towel with me in my equipment bag. I get them in places like Dollar Tree, Dollar General, Dollar Whatever, ... Some sort of terrycloth works best in terms of absorbing the drainage. Neutral colors (or black) work best in terms of not being noticed. I'm not swayed by that portion of the population which is so concerned about the hygenic issues. First, these are unavoidable. Second, what comes out of the instrument is (as others have pointed out) CONDENSATE -- i.e., condensed water vapor. I think that all the aversion and distaste that many (non-players, woodwind players, et al.) have comes largely from the historical mis-description of "spit valves".
If you're practicing in a school band room (as does one band I'm in), it really doesn't matter to anyone. If you practice on an auditorium stage (as in another band), then you have to realize that SOMEONE has to clean up whatever you dump on the stage surface. Regular cleanup crews are not enthusiastic about cleaning up what they (mistakenly or otherwise) regard as "spit". Maintaining good relations with one's hosts requires at least a modicum of consideration. It's really that simple.
As far as horns go, people should realize that there are consequences to not spinning and dumping. It's hard to beat the laws of physics. However, there is a range of technique involved in this that goes from "subdued or moderate" to "outrageous and ostentatious". In general, one should not perform an Olympic class horn spin in the context of a performance.
Gary Merrill
Wessex EEb Bass tuba (DW 3XL or 2XL)
Mack Brass Compensating Euph (DE N106, Euph J, J9 euph)
Amati Oval Euph (DE 104, Euph J, J6 euph)
1924 Buescher 3-valve Eb tuba (with std US receiver), Kelly 25
Schiller American Heritage 7B clone bass trombone (DE LB K/K10/112/14 Lexan, Brass Ark MV50R)
1947 Olds "Standard" trombone (Olds #3)