Paula , I was just in Istanbul , it seemed like a safe place to be. I know of a nice youth hostel ( hotel albion) 5 blks away from the grand bazzar( in the directon of the blue mousqe) and one blk off off Kucuk Ayasofya Street(turn right) . I went in there to ask prices and it was something like 30 euros? the cabbie from the airport will be very courtous and professional. enjoy your time there !
Approximate driving time is 25 minutes and distance is 17 kilometres (10.5 miles) from the airport.
Taxis are available from outside the terminal building. Airport buses run to central Istanbul from the arrivals terminal every 30 minutes.
Driving directions: From the airport, take the Seaside Road. Turn left at signs for Sultanahmet, drive through Kucuk Ayasofya Street, followed by Yerebatan Street and turn right onto Alaykosku Street.