Honestly, I don't trust most pet-grooming places.
I would recommend putting a towel at the bottom of the sink, and using a spray bottle or a faucet that can do a sort of shower spray. Your cat won't like it, but it might be a lot better than taking it to get groomed by a lot of strangers.
If you must do a groomer, check the place out first. Some places don't treat animals well... Looking it up on Google can help you in finding the reviews you need to see to make a good decision.