On the French side, just past the village of Grand Case (heading away from Philipsburg), there's a perfect little beach called (ta da!) Le Petit Plage. It isn't marked, so watch for the sign to the Grand Case Beach Club. Park along the road and walk down a smaller dirt road to the beach. It's a public beach, so no one at the hotel will mind giving you directions if you get lost.
This beach occasionally gets crowded, but is far less so than any other beach I've seen on St. Martin. There's a hotel bar and bathrooms you're free to use. Don't use the chairs and beach toys, though; they're reserved for adult guests. There's a little bit of decent snorkeling along the rocks.
If you get hungry, walk five minutes up the road into Grand Case, an adorable village and a true food mecca. There are restaurants from 5-star French bistros to shacks where you can get a wonderful conch chowder and a johnnycake for a couple of dollars.