Some more info:
The rates are $8.00 a night if you bring all your own equipment or $35 a night if they supply everything, including a tent (2 to 3 room), tarp, mattresses, sheets, dishes, cookstove, gas lantern, table, chairs, and water jug. Here are some pictures of a typical site:
The compound picture is taken from the water--the sites are right on the beach.

The bedroom picture is the view from lying in bed. Which we did a lot of. Note: they don't supply hammocks, so if you want one you need to bring your own.
The place isn't that well known--here's a picture of the beach taken at about 10AM.
As you can see, you don't need to share the beach with a thousand tourists. The building at the end of the beach is Nicole's beach bar--best pina coladas anywhere. Nicole also serves hotdogs and hamburgers for lunch and rents beach chairs. The beach has the best snorkeling of the entire island. So once you're there, you don't need to leave. I imagine that if you took your kids there, you wouldn't need to pack many clothes for the young ones because they could spend most of their time running around naked.
Of course, not even paradise is perfect--here's a picture of the women's bathroom:
It's rather grimy, even on third world standards. If you have to use the john at night, you might get to experience a two inch cockroach running across your toes. There's only cold water showers. But all in all, if you can stomach the bathrooms, you'll love your experience there.
Basically, Brewer's Bay is on the backside of beyond on Tortola. To get there, you will either need to fly in to Beef Island airport on Tortola and take a 45-50 minute taxi ride that will cost you $50 and scare the bejeebers out of you as you cross over the mountains, or else fly into St. Thomas and take a 45 minute taxi to Red Hook (at $30 or more), then a one hour ferry to West End on Tortola, then a 30 minute taxi ride that will really scare the bejeebers out of you. Once you get to the campground, you won't want to leave very often because getting over the mountains is time consuming and expensive. It's a good idea to take your own food (and diapers) because getting to the grocery store in Road Town, the capital, could easily turn into an all day event. There is only one restaurant in Brewer's Bay, and it's a pretty sorry place. (The campground does have barbecues once or twice a week that are worth signing up for to taste some local food.) But if you're content to just hang out on the beach and you've brought all the food you need, you'll be all set. Last time Seth and I went, we stayed 20 days and never left the beach once, except for the single day we went on a hike to the top of the overlooking mountains and back.
Weather is pretty predictable. From December through March it's in the '80's every day and the water temperatures are a delightful Caribbean warm. It might be a good idea to take a light blanket for the bed a night, but you certainly wouldn't need sleeping bags. When the moon is full, the water tends to get choppy. Then you can't really swim, but you can splash in the big waves. But in between choppy waters, the swimming is great. The idea is to go down for at least 10 days to two weeks. That way, you can rest up from the ordeal of getting there, and you can also be assured good swimming conditions in case you get a spell of 4-5 days of big waves. We've usually made our reservations for December or January in September or October with no problems. We've never seen the place get more than 60% filled by mid-January, although I guess they occasionally sell out.