Sharpening with damp stones is
not the water sport you imagine.
I work at home on my Steelcase desk with a small towel for rotating the different grit stones and another under the work area to absorb the few drips of runoff when I clear the loaded stone with the drip bottle provided. The desk is in my office and shows no sign of damage yet. A paper towel cleans up.
At the folks I worked on a lunch counter facing their kitchen. Same towels and minor dampness. Clean up is easy. Nobody, even my Mother-in-law, complained in the least about any mess in their fine home. Her husband even brought me dirt encrusted garden knives, along with every piece of cutlery in the house, from the garage to sharpen. I cleaned off the dirt first. Weeds are now in trouble.
People are facinated when they see the sharpener in use and want to talk to you about it while you are counting and concentrating using both hands at once with force on a sharp edge. Most people, I guess, are unable to sharpen anything by hand work and when they see it done with intent they are seeing something foreign, perhaps for the first time.
The pleasure my Father-in-law took from my work on his father's three Marbles hunting knives was very satisfying to me. They consisted of two skinners and a USMC tactical all with leather sheaths. I cleaned and polished the handles and blades. Then I restored the edges and mirror polished the new razor edges. He saddle soaped the sheaths. They had been out in the garage for decades. When I left, they were next to his chair, and he asked me not to move them as he wanted to enjoy them.
I don't bother to wear an apron or other protective gear. However, my dress is usually casual. I've never even put a stain on my clothing.
Best,
oregon