Kent County Council operates a volunteer warden scheme incorporating the North Downs Way, but in Surrey volunteers are generally managed by supervised groups such as the Ramblers Association (who are also very active in Kent)
Problems can range from missing waymark signs and broken stiles or gates to burnt out cars and flytipping.
You could also get involved in simple vegetation clearance and helping with visitor surveys, for which you'll receive the right training and equipment.
Useful links
Kent County Council: Countryside Access Warden scheme
Surrey County Council: Volunteering in Surrey's countryside.