Unfortunately, there are no campsites on Bear Mountain, and the Bear Mountain State Park doesn't allow visitors to camp there. Luckily, it's so close to the city that urbanites can come for a day trip. But if you want an overnight stay, there are other options.
For instance, Beaver Pond Campground is not far from Bear Mountain and includes tent and trailer sites. It even offers laundry and bathroom facilities with showers.
Bear Mountain Inn is situated within the state park at the base of the mountain and overlooking Hessian Lake. It's also an historic inn that dates back to 1915, and was even a training location for the Brooklyn Dodgers. The Inn has been recently renovated and reopened in 2012 after being closed for years.