Cozumel-stay in your groups!!!
Grand Cayman--no worries there, very safe and clean
Ocho Rios Jamaica-the beauty of Ocho Rios is when you get on/off the bus you are in a gated area and if you want to go do some shopping you don't have to walk too far. Island Village shopping is about a block walk up the street from where they pick up/drop off for tours. It took under 10 minutes to walk from the end of the pier to Island Village. Yes, the cab drivers will ask you if you want to go somewhere, but if you are polite they leave you alone. When we went in and left Island Village we were asked if we needed a cab and politely said no we'll walk back to the ship. The police/security in the area of the port and shopping is quite good, they make sure the passengers are not harrassed.