eb- I back up what Dave said. Both Cozumel and Grand Caymen are safe and even if you have to get back on your own (say something happens to the tour bus where it breaks down or something) both are small enough and have plenty of Taxis where you can get back to the ship without a problem.
The only exclusion to that would be if you tried to do a tour on the mainland Mexico from Cozumel. I have been there plenty of times and would still be concerned trying to do something like that as you have to do a ferry trip to get there.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007