I'm trying to think... I lived in or around the city for six years, but never had to stay in a hotel.
I know where the absolute cheapest, non-infested, metro-located hotel is on the VA side of the river... it's an EconoLodge in East Falls church, near I66.
If you stay *in* DC proper, Georgetown is technically not metro-accessible, although there are busses and it is not a bad walk from Foggy Bottom. Foggy Bottom doesn't have very many majors hotel chains right near the metro.
To be honest, I'm more familiar with the VA side of things. Rosslyn has PLENTY of hotels within easy walking distance of the metro. It's technically North Arlington, but anything that advertises themselves as "Rosslyn" should be an easy walk to the metro. I think some advertise themselves as "Key Bridge" as well.
Further south, closer to DCA, there is nothing *on* the DCA metro stop. It basically serves the airport only. For that part of town, the best bet is probably Crystal City. It's actually S. Arlington, but anything advertising itself as "Crystal City" should fit your bill.
The two areas are on separate metro lines, but the way the lines run, you will be very close to the things you want to do. If you stay out in MD (I don't suggest it) you'll have a much longer ride, plus you'll have to transfer lines.
If you're looking for a specific hotel, sorry I didn't name one. The Rosslyn area has enough of the major chains that you won't have a problem. There are PLENTY of hotels within a 4-block radius of Rosslyn for sure. It's probably cheaper than downtown and just as convenient.