Sultanahmet puts you steps from Hagia Sophia and the Blue Mosque but is very touristy and quieter at night. Beyoglu (Taksim/Galata/Cihangir) has far better restaurants, bars, and nightlife — and is where most younger travelers and expats base themselves. Kadikoy on the Asian side is authentic and affordable.
Boutique hotels in restored Ottoman mansions in Sultanahmet or Beyoglu are Istanbul's distinctive accommodation option. Cihangir neighborhood boutique hotels suit those wanting a local residential feel. The tram and metro connect Sultanahmet and Beyoglu easily — location matters less than it appears on a map.