I just left Istanbul. As others mentioned, Taksim Square is heavily populated even at this late hour and is a relatively safe place.

What one could do is take the metro from the airport to Zeytinburnu and change to the redline for Kabatas. Then, take the funicular to Taksim Square. This can be done in about an hour and will cost 5.25 Turkish Lira.

If the tram stops early for some reason, say midnight, then you can take a taxi from Kabatas to the Grand Hyatt. This would be only a 0.25 miles at most but is all up hill. You wouldn't want to walk it with luggage.

Enjoy Istanbul.