This might be a time when it's worth the effort to register with the USA Department of State (website now, so you apparently don't have to contact the local consulate any more). It might make you and wife feel better. If you're concerned, either stay in a low key and more local hotel or stay in a top place with good security. [I've been impressed by the security at the Cirigan (spelling?) Palace, but I haven't been in Istanbul for a while.]

I would say the biggest risk is people (especially kids) asking for money around the tourist sites.