St. Thomas--If you dock in Havensight, walk toward the entrance to the cable cars (toward your left and ahead as you get off the ship. Just before the place where you get the tickets for the cable cars, there is an oriental restaurant. Wifi is $3 for all day, cheaper if you buy a meal. Lots of crew there, busy, safe place. This is less than a 10 minute walk from the dock...maybe less than 5.
There is a place in Havensight Mall but it is in a bar and there was no one there when we went past--it just didn't feel safe, so we kept walking.