I spent a week there a while back, without a vehicle of any kind, and I was impressed with how law abiding the central old part of the city appears to be, both day and night. The back streets in the old area are very narrow though and delivery trucks, vans etc just park anywhere to offload. Once they are in position they don't move for anything or anyone.
If you search online for specific hotels with garage facilities I guess you will find one or more, but the bigger international chains are more likely to have room for vehicles. In general they are located out of the old city centre and some of them have good views of the sea of Mamara. A lot of hotels will give a discount on their stated rates for payment in cash and the Euro is widely accepted.
Hope this helps. Enjoy Istanbul, it is a very lively city and not at all intimidating.