I came on here to recommend Avon Tyrell Activity Centre where we just spent the weekend. However, it isn't really walking distance from anywhere I don't think - not easy walking distance anyway. It is near to Bransgore and is certainly easily drive-able from good pubs etc.
We had a fantastic weekend and there are lots of outdoor activities to choose from for kids and adults, though you need to book them up fairly quickly. In our case, the children did bushcraft and high ropes, but there was canoeing on the site's lake, a zip wire, a climbing wall, archery and I think other activities too. Also, a heated swimming pool (only open at certain times when we were there but it may be open more in the summer).
You could chose to camp with or without electricity and the loos and showers were pleasant and cleaned regularly I think. You could also reserve a free barrel BBQ or have a fire in your own raised firepit, as we did (many sites don't allow any type of fire in the New Forest..
Where we camped, there was also a children's playground which the children really enjoyed and made friends there too.
There is a cafe on the site which, again, I suspect is open longer hours and with a bigger food selection in the summer.