There have been many studies done which show people are far more likely to wash their hands in a public restroom when others are present. Putting an attendant in each restroom would take a lot of staff but it might help. When a friendly attendant comes over to the sink and lays a towel there for you to use, it is hard to pass on by! The attendant could also use gloves to open the door.
However, what about the people who simply use the bathroom in their cabin? I would suspect that the rate of hand washing is much lower there in any case.