First, you can stay on the ship. THere is no adult programming, but the kid activities continue. You can hang out at the pool, etc. Alternatively, go off the ship but stay in the general local of the port and eat on the ship. There is a shopping mall at the dock at St. Thomas.
There is a beach by Coral WOrld on St. Thomas. You can take a cab to Coral World, but I've been told that free transportation is available to the beach (Not sure how as I've never done it). If you want to do Coral WOrld, you can then walk across the street. There are discount coupons available at the tourist agency at the dock. This is a "swimsuit optional" beach.
The best way to save $ with a 10 year old is to do any excursions on your own.
DCL calls a 10 year old an "adult" for pricing purposes, but most other people start adult rates at 12 or 13. This can make a huge difference! Almost any excursion that can be booked thru DCL can be done on your own at a savings. There are a couple exceptions where DCL has booked the whole place for a day, but these are rare. Another advantage to going on your own is that you miss the crowds.
Buy or borrow(from friends at home) your own snorkel equipment--snorkeling at Castaway is free, but you need to have equipment. Buying it is about the same cost as a day's rental. Alternatively, rent one set and the adults can take turns.....snorkel vests are required but are free. You can always exchange flippers if needed for size reasons.