Cozumel is easy, and safe, but you should count on taking a cab to get anywhere, unless you just want to shop. There is no sightseeing in the immediate pier area. Drives can be 10-15 minutes, or further, depends on what you want to do.
Cayman is easy and safe, too, thanks to Disney's pier being right in the heart of Georgetown. You can wander around there, or take a mini-van-taxi to the beach (5-15 minutes), etc.
It IS POSSIBLE to walk to the beach from the downtown pier, have done it, but it's quite a walk for most.
In Falmouth, I would not leave the gated cruise shopping area, unless on a guided tour.
I'm saying this as someone who has left the pier area on my own in Honduras...to hire a driver to go to some of the reefs... but now, and with kids, no way.
You have to drive to get from Falmouth to any of the attractions, can be up to a 40 minute drive I believe.