Yes...all you've heard is unfortunately true. But I live here, and that makes me fit to help you avoid becoming a victim of any kind of crime. It's real easy to be safe.
Here's the scoop: everybody, from the lower middle classes to the high ones, have a car. So stepping on the public transportation will instantly introduce you to the very poor, and the petty criminals. There you are, at least a head taller than the crowd, carrying around a map and a camera... guess who's the easy target here. I don't recommend renting a car, because if you don't know your way around, you'll easily be trapped in the worst traffic jams you've seen and maybe get lost in dangerous parts of town. Some of them are right next-door to the touristy places.
Ride 'taxis de sitio.' These are the cabs you call for. Your hotel or host should be able to give you a phone number for the closest taxis site, and they'll come pick you up, sometimes agreeing to get you again wherever they left you, at a designated time, or otherwise referring you to another site. Do not hail cabs off the street, you might get mugged. If you do get mugged, for whatever's holy to you, give them the damn bag. They'll probably just snatch it and run. If you fight them, you'll get into more than you bargained for. So keep little cash in that bag, conceal the passport and big money somewhere inside your clothing close to your body or in the hotel's safe deposit.
The subway is a rather good one. I recommend the line from Polanco to the Centro (downtown), but many others aren't really safe. Avoid it at night, or just be on your toes.
Keep handbags crossed from your shoulder across your chest and have a firm grasp on it. Do not get distracted. Keep alert and watchful of people approaching you or passing you by. When in doubt, follow your instinct. If you get scared, don't risk it. Keep off dark places, avoid straying in lonesome ones too. Look self-assured. You'll be all right.