Phoebe does not know how to lick her chops after she drinks, I swear. Consequently, my husband and I, and other guests, have more than once slipped in the river that she leaves beside her water dish. I have to wipe it up periodically everyday. Has anyone ever trained a dog how to lick his/her choppies?
We have a marine made mat that sits under her watering tank. This cuts down on the river and soaks up most run off from her chin. The reason I use a marine mat (boat mat) is because when it gets wet, it doesnt allow the water to get to the floor, and it cant stain your floors like some mats. They are designed to hold the water in the mat.
I think it would be far easier to put a towel or mat under the drinking bowl than to put it on cue. But yes, you probably could do it by capturing the behavior using a clicker. You'd have to be ready for a time when she's likely to do it (e.g., you could put a dab of peanut butter on her nose), and when she licks, *click* and treat. (This assumes that she's conditioned to the clicker.) Click and treat, click and treat. After a number of repetitions, you could add a command ("Wipe your face!"). Eventually, you would fade the treat and the click.