I have always been a little leery of cab drivers in Istanbul, especially around Sultanahmet. I have had three experiences with taxis in Istanbul. The first time, our tour guide was hesitant for me to catch a cab and suggested that I instead take the streetcar to a taxi stand outside the because the cab drivers in Sultanahmet tend to rip off tourists. The second time, I only had to go a couple blocks, maybe 1/4 mile, through a tunnel that was not accessible to pedestrians, and the cab driver charged me about $16. I was in a hurry, and didn't negotiate as I should have. The third time was okay, but that was only because our tour guide negotiated everything up front. (The reason I had to keep catching cabs on a tour was because my traveling partner was ill, and I was dealing with trying to get him to hospitals and doctors appointments and such.)
Check and see if your hotel can provide transportation. Many hotels, even the small ones, offer airport transfers. This method has always worked for me.
By the way, Ataturk is the old airport. It only handles cargo these days. The new airport is Istanbul International, which opened last spring.