I park near Claiborne and Earhart for Saints games. It is not a good spot if you are looking to walk around downtown and the French Quarter before and after the game. But if you plan on driving to other parts of the city after the game that is where I would suggest. Parking is free there and you avoid downtown traffic.
You can park on the street on Earhart or underneath the interstate, but if you do park underneath the interstate make sure you pull up far enough so you're not blocking the sidewalk (they will give out tickets, trust me I know). Park and take the stairs over the railroad tracks and to the Dome.
Here's an address you can put into your phone - 2929 Earhart Blvd, New Orleans, LA 70125