How safe are Cozumel, Mexico & Montego Bay Jamaica?

I am going on a cruise that ports in Cozumel Mexico, The Cayman Islands and Montego Bay Jamaica and I was wondering how safe these locations are? I was also wondering what we should do once we get to these locations the cruise line offers a lot of excursions but they are kind of pricey, is it ok to go do things on your own?