What is the safest area to stay in New Orleans?

Kenner near the airport, but some parts near of West Metairie are urban war zones at night. Avoid most of the motels along Airline Hwy.

The best neighborhood in New Orleans is Lakeview.

Regarding everywhere else, if you are visiting as a tourist then all you need do to be safe is use basic common sense. "Basic common sense" includes not buying, selling, or using illegal drugs or associating with people who do.

Pick a place that's above sea level.

Generally around the French Quarter is safe, but there is a lot of purse snatchings and pick-pockets on Bourbon Street in the evening when the streets are crowded. Word to the wise....don't over drink. You become a target. Other advise, don't engage with any strangers. They are either scammers or thieves.