The site I use when I go to the New Forest is Red Shoot Camping Park.
A privately run site with a toilet and shower block, bike hire on site(I think) and a small shop. There is a cattle grid as you go in to the site so the Horses cant get in.
There is also a lovely pub right next door (actually you drive though the pub car park to get on to the site)
http://www.redshoot-campingpark.com/