As a general principle, I wouldn't classify Istanbul as a particularly unsafe city.

Sultanahmet area would be my suggestion for a first-time visitor too: many of the "must-see" sights are in that area or within easy walking distance, and there are many, many hotels around there to suit just about all tastes and pockets.

Some people like to stay near Taksim Square, but that, to my mind, is not nearly such an interesting area (although İstiklal Caddesi is good for shopping, western-style, and eating). Others enjoy staying in the Beşiktaş/Bebek area or other localities along the shore of the Bosphorus, but really I think you can't go past the Sultanahmet area.