I've sat all over the Superdome over the years, even before Hurricane Katrina and the renovations. The upper levels are less than stellar. Any seat you can find in the "500s" will be significantly better than the "600s" both in terms of sight line and overall experience. 651 was where I sat most recently for the Carolina Panthers game in December. The Saints won big time and the Panthers turned out to be a tough division rival so safe to say, that was one of the more exciting games I've seen in this stadium.
651 will put you in the upper north end zone. Not the greatest seats in the world but good enough for fans to be able to tell what' going on and a awesome location to watch the touchdowns scored in the north endzone. The rows are numbered 2 through 21. Stay as close to the bottom as possible to avoid the stair climbing because those stairs are as steep as I've seen. Surprisingly enough, the ushers were checking tickets even on the upper level. I don't know of too many fans that would intentionally try to "sneak" into these seats but I guess it does happen.
The Superdome is one of the most visible landmarks in the downtown New Orleans skyline so if you visit New Orleans without coming to a Saints game or concert here, you'll want and need to make a 2nd visit. Make a point to check out "Champions Square" before a Saints game. It's where all of the live music, drinking and partying is taking place.