DC Hotels are going to expensive... You can actually stay in the Falls Church or Vienna areas and use public transportation... Unfortunately, the subway doesn't go to Alexandria but the buses do so you may be able to use buses exclusively...

Alexandria is fairly safe and better closer to the Potomac than out a ways...

You might also check on the MARC trains to see if they stop at Alexandria... The train station used to be right across from the Masonic Temple... Not sure these days if MARC stops there but if it does then you could stay all the way out in Harpers Ferry, West Virginia and train in from there...

B~