While all three ports are safe enough to do your own excursions, I wouldn't go too far off the beaten path in Nassau or St. Thomas. Both places have problems with criminal activity if you happen to wander into the wrong place.
St. Maarten is a good place for going to a beach since they are plentiful there and easy to get to. There is a pretty nice one near the cruise ship pier. Or perhaps take a cab to the beach at the approach to the airport. Google 'St. Maarten airport' and you'll see what I mean.
For a more quiet time at some nice beaches, don't stay on St. Thomas but instead go to St. John. You can take a ferry to it.