We stayed at Fredericksburg KOA They will take you to the train into DC and pick you up. Or youyou can drive 45 minutes up I95 and catch the metro. We did both and this kept me from driving in and around DC
I have stayed at Brunswick Family Campground in Brunswick MD. It is a short 1/2 mile drive to the Metro. Not sure how long a drive into Washington. At the campground, you will park right beside the Potomac river and it is reasonably priced.
There are also some campgrounds in Fairfax, VA area as well. Cherry Hill is one place, but one of the most expensive RV parks that we know of; however they're clean and well maintained. You could try Greenbelt NP - then drive to the Metro in Greenbelt and ride that into town. The metro map (website: http://www.wmata.com/) can help you plan your trip from Greenbelt Metro to areas in DC. Enjoy the sights - just remember - they could be blooming early because the weather have been on a different cycle compare to the last few years. They're wonderful!
We have used Aquia Pines - Stafford, VA every time we go to Washington. You can get a shuttle from the campgrounds - or for $5 and parking - use the Metro - two exits up 95. The subway brings you to the middle of the "mall" near the Smithsonian. The park is woodsy (yet close to shopping centers) and clean. Great place! http://www.aquiapines.com/
Pohick Regional Park, a public park that is just minutes from the south end of the metro line. Large timbered sites with water & electric, plus a dump station. Cost was $18/night when we were there two years ago.