I've stayed at the Red Roof Inn Hotel near Verizon Center in Chinatown. The rates are reasonable, it's close to a lot of stuff, and you can take the Metro to RFK via the "Gallery Pl-Chinatown" station. Take the yellow or green line south to the blue or orange line at L'Enfant Plaza. Go east on the blue or orange line to "Stadium-Armory" and follow the crowd.
Red Roof does charge for parking, but it's under the hotel and reasonable, added on to your bill.
I'LL PRESS THE CAPS LOCK AND SAY THAT DC IS NOT THAT BAD. THERE ARE A LOT OF HOMELESS PEOPLE DOWNTOWN, MAYBE EVEN MORE THAN OTHER BIG CITIES. DON'T LET THAT SCARE YOU, ESPECIALLY AT THE STADIUM-ARMORY STATION. JUST FOLLOW THE CROWD AND YOU'RE FINE. OTHER THAN THAT, THE DC METRO SYSTEM IS ONE OF THE CLEANEST AND SAFEST IN THE COUNTRY.
Just act the same as you would in any big US city, you'll be fine. PM me if you need any more info about the area.