Not really sure what you are looking for that is "Tropical" but if looking for green rain forest type of terrain you can jump over to Saba and get right into the rain forest pretty easy. If staying at Queens Garden there is a trail that starts from across the street that you can hike over to the main trail that gets you to the top of Mount Scenery. It is a tough hike, basically a goat trail. There are no beaches, but it is a nice relaxing break from SXM. Anguilla is the opposite, nice beaches and little vegetation but I would also call that tropical.

Cheers,
Todd