Try Hollands wood. It's a forestry commission site. You can take dogs on it and the facilities are clean and plentiful. The only thing is it is quite large but probably the best site for a non-camper. Plenty to do nearby.
I've never checked if you can take dogs into the forestry commission campsites, but they are lovely. I think the website is something like campingintheforest.co.uk. There are several in the new forest.