Where to stay near/in St. Tropez?

At what time of the year? Traffic is horrendous in summer, but there are boats from Ste Maxime, which would be a good summer base.

Oh- right. June 20-25

Ste Maxime is great. The hills in the Ramatuelle/Gassin village area are also good, with a quick drive to the beaches near Pampelonne.

Luckily you are traveling in June before the school holidays in France begin on July 7.

St-Tropez has no train station, so it's not as convenient to reach as the other places along the coast. The regional train line makes sightseeing in coastal towns very convenient. I would think long and hard about choosing a base without that benefit unless my main interest was to visit towns and villages off the coast, up in the hills. With a family of six, you may be looking for a villa experience, with a car, and that would be a different situation--but day trips to coastal towns would get annoying, I'd imagine, between traffic and parking challenges (and costs).

May I just underline the summer traffic situation at St. Tropez - unbelievably heavy with long backups. There is a reason there is helicopter service from Toulon, Cannes and other points farther to the east, as well as sea shuttles from Sainte Maxime. Consider staying in Sainte Maxime and visiting Staint Tropez by boat.