How can I keep my surfing booties from stinking? Also, what is the best way to dry them?

Wash ur feet more often! No just kidding, well no that might be a factor, l do not know? You do need to wash them out with fresh water then turn them inside out to dry or place them in an good air flow like an air conditioner vent where drying does not take days! You can use a mild soap when washing like dish cleaning soap, just rinse real good. When l was diving offshore we had vents on the vessels that blew warm air, these were perfect for quick drying. You can use a dryer but too long & they tend to shrink a bit & some dryers might damage the bootie if they get caught during the process. Probable tumble dry on low or no heat may be just fine. Another thought, if u r wissing in ur suit & u probably are, this may be part of the stink, so washing really is a good idea in that case.

Why bother drying them. .... strange

You can not. Sorry but the truth hurts, or in this case stinks. You can try washing them out after very use and making sure they dry quickly but in the end they will still stink.