I was so worried about this with my 3 year old. This was my experience:
He had been out of diapers except for sleeping for almost 11 months when we cruised, but still got a lot of help in the bathroom from me or a family member. My outgoing child had no problem whatsoever using the potty alone, and all of my concerns were unfounded. The bathrooms in the kids' clubs are manageable in size for children and he was able to get on the potty without help, as well as wash without help. If your child does have problems, the staff will call you on your wave phone and tell you. They do supervise but won't touch your child. The cruise actually spurred him to become more independent at home!