By the time you get out of the airport and travel to the Blue Mosque in rush hour, it might be close -- but the sun sets pretty late in July/August, so you should make it.

Safety is not an issue in the main tourist areas (though if you stray onto the streets off of the main boulevard that passes from the Blue Mosque past the Grand Bazaar, or in the back streets on Eminonu, it can get dodgy; I wouldn't be worried, just take the same precaution that you would in any large city).

Fastest way would be to tell a taxi to take you to the Blue Mosque from the airport. With no traffic, it should be less than half an hour, but it could conceivably take several hours if you get unlucky. There are trains, but it would take two longish rides -- transfer from the Metro to the street car/tram at Zeytinburnu, though it's a bit of walk and painful with luggage, then get off the tram at Sultanahmet.