Being from the DC area and having just graduated from GW, I can give you my $0.02 on living here:
1. PLEASE avoid going for any of those cheaper apartments in NE/SE. This is especially true if you are a female and will be walking/taking the bus. I cannot stress how dangerous it is in this city (I'm not being paranoid or exaggerating, this is coming from a tough chick married to a Marine martial arts instructor and knows a thing or two).
2. Don't assume that ANYWHERE in DC is "safe." I lived in Dupont Circle for a couple years in undergrad (also at GW) and there were multiple armed assaults and rapes of women -- inside their supposedly secure apartment buildings. It was nice to live in a such a cool neighborhood and walk to class, but I still carried pepper spray and a pocket knife in my purse and looked over my shoulder constantly.
If you can, I'd go with some of the above suggestions. I've tried them all, literally: I grew up in Silver Spring and kept a job there in undergrad, I commuted the first year of undergrad from Gaithersburg, MD to DC everyday (spending either 2-3 hrs a day in traffic or on the Metro); I lived in Dupont Circle and walked to school; in the last few years I've lived in Alexandria, Annandale, and Oakton/Vienna and mostly relied on driving to the nearest metro station and riding it for 10-30 min each way.
Your best bet for cost, quality of housing, and safety would be in a neighborhood located on the metro line in Northern VA (I'd suggest Clarendon, Courthouse, or Rosslyn if you don't have a car). Or, if you really plan on spending a lot of time on campus to study or are into the downtown club scene, lots of people in our class enjoyed living in Foggy Bottom. Georgetown has some nice places to live, but is not on the metro and can cause a hassle.
I've just moved to the complete opposite of DC (Charleston, WV) for my residency and I know I made the right choice: no traffic, polite and friendly people (didn't know they existed!), plenty of outdoor pursuits, beautiful scenery, and a 2500 sq foot house for $950/mo that's 3 minutes from the hospital. It's the greatest culture shock I think I'll ever experience!
Good luck in school. If you have any questions about stuff to do and see around town, or about GW, feel free to PM me 😉