The old Beaulieu airfield is out as this is now part of the new Forest and has walkers and animals all over it. Also, much of the hard surfase has been pulled up and it is very uneven. There is a very short strip that is used for model aircraft flying.
The Lymington strip is privately owned. It now has an 'X' on it. The owner is unlikely to let you use it. It have never seen anything (apart from one of his tanks !) on it.
There is a strip near Sway (west of Lymington) that has a hangar and is occasionally used. I do not know any details. Plenty of hotels to land a helicopter in the area.