No train, but there is a bus. It's really cheap, but can be a hassle if you travel on a weekend.
The bus is called the E-2.
http://www.jeffersontransit.org/schedules/E2Airport-Map.htm You pick it up in the middle of the median outside the arrivals ground transportation area. After you exit the sliding glass doors, you cross the street halfway to a bus median. In the middle of the median is a bench and a small sign that says E-2.
I think the bus runs every 1/2 or so and it costs $1.60 per person with no change returned if you don't have exact change.
It drops you off at the northern part of the Central Business District (Canal & Saratoga) near the Tulane Medical Center. It's about a 1/2 mile walk to the French Quarter.
On weekends, the E-2 only goes to Clairborne & Tulane Ave. in Mid City, which isn't the safest of neighborhoods. There you walk a 1/2 block south on Tulane Ave. and take the 39 bus all the way down Tulane Ave. and it ends at Canal & Saratoga in the Central Business District.
Unless you are familiar with New Orleans or are particularly adventurous, I would recommend just taking a cab. The E-2 is mostly used by airport employees, college students, and thrifty locals.