The Northwest Area is safe. When you get up around the Capitol it's iffy. And no matter where you are, don't walk at night or take metro...take a cab. I lived in Adams Morgan for a year. I found it perfectly fine during the day, but at night it became another place althogether.
How about the southeast? I lived on B.A.F.B. for 4 years in the 60s. Have the gangs taken that over yet?