In general, it is possible to find some kind of accommodation in any district of Istanbul. Here is a quick list of the districts where they are concentrated most:
Harbiye is a popular place to stay, as in the main center of the new city on the European side, and contains a variety of international standard apartments, hotels, and moderate hotels for budget travelers. Nişantaşı and Taksim are 5 minutes from Harbiye so you can stay in Harbiye and benefit from all activities in Nişantaşı and Taksim.
Taksim is the main center of the new city on the European side. Locals and tourists go to Taksim for shopping and entertainment, as well as moderate hotels for budget travelers. There are also two hostels in this area.
Sultanahmet the main center for the old city on the European side. It has a selection of quality, reasonably priced hotels, many with terraces overlooking the Golden Horn, or with views of the Marmara Sea and the Blue Mosque. Most hostel-type accommodation frequented by independent travellers are located in this district, although it is possible to find a few upmarket hotels.
Quite pricey hotels can be found in western suburbs, especially around the airport, as well as on/overlooking the banks of Bosphorus.
caravan
With the closure of relatively central Ataköy caravan park, the place where you can tow your caravan nearest to the city is now located in Selimpaşa, a far outer western suburb of the city, though it is still a good 40 km away from central parts of the city.