Another vote for Sultanahmet area. We (wife, teenager and myself) stayed at the Hanedan Hotel, which was a nice clean budget hotel (not far from the above mentioned Four Seasons). Had a family room for a reasonable rate, no frills but clean and comfy, I'd stay again. There are many other options, too. The area, and in fact most areas we went to in the city, felt as safe as any other city in the world, no worries.