CLIMB V.I. charters boats out of Tortola, British Virgin Islands. There is no commercial air service from mainland U.S. to Tortola's international airport, although a number of small airlines fly in from Puerto Rico.
A cheaper option is to fly to St. Thomas, U.S. Virgin Islands, and then take the fast ferry to Tortola.
You will need a passport to enter the B.V.I. from the U.S.