I can't speak about Montego Bay. But Cozumel and Grand Cayman is what I consider safe.
You can't walk to Paradise Beach, you have to take a taxi. And to go to downtown Cozumel you have to take a taxi.
Paradise Beach is free, if you want a chair or shade just buy a drink or some food. Find Tom and tell him Dwayne sent you
There is a neat snorkeling place close to ship tenders (I know I have posted many times) on Grand Cayman within walking distance. You will be amazed at the different islands and cultures.