İstanbul is undoubtedly far safer than most European or North American cities, and cases of mugging and assault against tourists are rare. Having said this, the crime rate is soaring, due in part to the increasing disparity between rich and poor.
For the average visitor, pickpocketing is the main cause for concern: be particularly careful around Sirkeci station, the Eminönü waterfront, the Galata Bridge, and around Taksim (especially at night). Also avoid being on or around the Byzantine land walls at dusk/night. Be very careful, too, on public transport, particularly when it is crowded. If you feel anyone is harassing or attempting to pickpocket you, try calling out imdat!, ‘meaning “help!”,’ and contact the tourist police.