DC is a pretty pet-friendly place, but you probably wouldn't get that from hanging around the typical "touristy" areas like the mall. In the residential areas and in most neighborhoods, you'll see plenty of people walking dogs. Downtown, the National Mall, and Georgetown (the tourist hotspots)? Not so much.
I don't own a dog (I have 2 cats), but I see enough people who do to assume it's pet-friendly. There is one fenced dog park I know of at the congressional cemetary, and I know there's a lot of push to add more.
MetroPets Yellow Pages: Dog Parks