I’ve lived here all my life, and you’re right about that radius around the most predominant tourist sites.
Let me revise that a bit: Georgetown is pretty safe, and the K Street corridor and business district is safe during the day (I routinely jog after work in the business district and have never had any trouble, except, perhaps, for dodging a few attorneys with briefcases trying to hail a cab!). I lived in Adams Morgan for a year — nice, historic apartment — saw a bust almost every night from my window. It was, however, a great location for telling people about the Lord! I was amazed at how receptive people could be.