I moved to New Orleans about two years ago. My friend moved here about a week after me. I moved close to the CBD, a couple blocks from the Superdome and in the heart of the city. My friend moved to Metairie (the outskirts of the city) in a gated apartment complex because she "wanted to feel safe".
Since we have been here, she has had her purse stolen, her car hit twice, and been mugged while in the Quarters. I have had not one problem since I have been here.
The truth of the matter is that if you are aware of your surroundings and practice safety at all times (like walking into one of the many well-lit bars or stores if you feel you're being followed in the Quarters, or not leaving your purse or belongings unattended), you should be ok.
Of course, things happen everywhere. Other than that, it's trulyabout being smart; the same as it would be anywhere else.