Don't know of any RV parks with BART in walking distance. The Fair Park in Pleasanton is close to the Dublin/Pleasanton BART station. Perhaps there is a shuttle or bus from the RV park.
Trying to drive to the BART stations weekdays is a challenge. Most of the parking lots fill by 8am. It is possible to reserve a spot online at some of the lots ahead of time.
On our last trip we went ahead and drove into SF. Kids wanted to drive across the bridges anyways. We parked in the open lot on fishermans wharf, then road the electric buses or cable cars around town.