RCI provides towels for Labadee. You can get large green towels at the Solarium at other times. The Explorer is a great ship and I had the time of my life last year. Enjoy! I wish I were leaving tomorrow.
The ship gives you towels for private island excursions, perhaps Labadee. The ship has towels in your cabin but they are not large enough for the beach. You may want to bring your own or buy a souvenir towel in the shops onboard. I'll be on the ship also. If I can recommend a beach in St. Thomas it would be Magen's Beach or Little Magen's beach for nude bathing. While in St. Thomas you can also take a ferry out to St. John. The beaches there are a bit more unspoiled and secluded. You can check out any of the beaches on St. John. Nude you can go to Salomon Beach right off the ferry docks. In Nassau check out Paradise Island. You can catch a taxi over to the Atlantis resort. The beach there seems to be the best. You can also check out Cable Beach where "Jaws" was partly filmed. San Juan... can't help you with. Try some Puerto Rican food is all I can recommend. You may want to see the Fort near the docks.
Yes, you will get towels, don't waste the suitcase space on blankets or towels. They have towels down by the pool and if you are leaving on shore excersions, they have them right by the gangway, and you can also drop off the wet towels there when you come back.