Just got back from Fairbanks and the best RV park in proximity to downtown, the airport, UAF and its wonderful Museum of the North, and, most importantly, Fred Meyer is the Rivers Edge Campground. It is just off the Parks Highway near the east end of the Chena River bridge.
We stayed at the Riverview RV Park on Badger Road. It is also on the Chena River but nowhere near downtown. It is located way east of Fort Wainwright and several miles north of the Alaska/Richardson Highway. If we had been with kids, we would not have stayed that far out of town.