The British Virgin Islands are a mere stone's throw away from St. Thomas and St. John, and accessible by ferry or private boat. However, you will need a valid U.S. passport to visit the BVI.
Take a "Closed Loop" Cruise
You can still cruise to the Caribbean without a U.S. passport if you are a U.S. citizen, but only if you take what is known as a "closed loop" cruise. That means that your cruise ship needs to start and end at the same U.S. port. The good news is that most cruises originating in the U.S. operate as closed loops (the exception would be something like a Panama Canal cruise that starts in Miami, for example, and ends in San Diego).
However, there are a couple of caveats. Some Caribbean countries -- Barbados, Guadeloupe, Haiti, Martinique, St. Barts, St. Martin (but not Dutch St. Maarten), and Trinidad & Tobago -- will require you to have a passport to enter or exit. Always check with your cruise line first to see if this applies to any of your ports of call unless you want to be stuck on the ship. Also, if something goes wrong with your cruise and you have to fly home, not having a passport will be a problem.
If you're taking a closed-loop cruise without a passport you'll need proof of citizenship and, if you are over age 16, a government-issued photo ID. But again, your best and safest route is to spend the money to get a passport before you travel.
Get a Passport Card
Think of a U.S. Passport Card as something falling between a Passport and a government-issued photo ID. It costs half the price of a passport, but can only be used for land and sea entry into the U.S. from Canada, Bermuda, the Caribbean, and Mexico. It cannot be used for air travel.
Practically speaking, that makes it not much more useful than a driver's license for Caribbean travel. Technically speaking, you could use it to cross the Mexican border and drive to the Riviera Maya. But that's 1,400 miles each way, so we're pretty sure you'd rather get the passport and book a flight, instead.