Istanbul is a LOT safer than any comparably sized city in almost every respect. Still, with 16 million people, there's bound to be a few bad apples. Many women report being harassed in taxis, on crowded public transportation, or on Istiklal Cd. Although you shouldn't let those experiences scare you off, you should take a few steps to keep safe.
#1, When possible, don't go out alone- go with a hostel friend or whatever.
#2, If you take a taxi, sit in the back seat, as the front may be interpreted as an invitation
#3, If you are in any situation you feel uncomfortable in, don't be afraid to make a big fuss- not only will this embarass potential harassers, but it will galvanize bystanders into helping you.
#4, Make sure you appear confident and refuse to engage or look upset at any verbal abuse or catcalls.
again, these situations are highly unlikely, but it's always best to be careful. For more information on Street Harassment and safety in Istanbul, please check out http://istanbul-en.ihollaback.org/
As a woman who lived in Istanbul for three years, I would say I felt completely safe in virutally every neighborhood. I wouldn't let concern about being in a 'Muslim' country stop you from traveling there. In fact, I generally felt safer and less harassed then I did in western Europe and the US.
You don't need to change how you dress in general, but keep in mind that you'll have to cover your arms, legs, and head to visit any of Istanbul's gorgeous mosques. Most Turkish women don't dress any more conservatively than your standard New Yorker.
Use your common sense. Don't go to strangers' houses without company (because Turkish people are so hospitable that they probably will invite you to a home cooked meal upon first meeting you). Don't get so drunk that you can't take care of yourself. Remember you're in a foreign country, but also remember that people are on the whole kind and Turks probably more than the norm.
