The NFL is cyclical. While no fan base wants to think about it, sooner or later a team is going to have a down cycle. The salary cap and draft are designed to make it happen.

I don't think the Saints are up for one as long as Brees is here. And depending how long it takes to find a quality QB and good young talent, the dip may or may not happen when he's gone- but somewhere in the future, the odds are the Saints will have a string of years where they aren't expected to contend.

But to answer the OP, I don't know about you personally, but the diehard fans will remain fans. A lot of the bandwagoners who may have jumped on ship in the years preceeding or just after the Super Bowl may fall away, but the die hard fans will remain.

With the cap rules, the league is really designed to be mediocre. The NFL calls it competitive balance, but frankly it's pretty watered down. A lot of teams go through cycles because they aren't very gifted in the art of foresight, while other organizations just aren't good at anything (Raiders, Jaguars, Browns, etc). But there are a couple of teams that have been good at managing the cap and sustaining success (Patriots, Colts). The Saints and Steelers are very similar over the last decade (decade plus, for Pitt) in how they fluctuate between being contenders and just being average, but they don't completely fall apart.

The Saints did well for a while with finding hidden gems that allowed them to keep the cost down and retain a certain core of players. But once guys started looking for a big pay day we over-payed on some players, and it's bit us in recent years. The offense keeps us competitive but we really haven't been able to keep up with the elite.

It really all boils down to finding the right QB when you make a transition, and building around him. Just look how quickly the Colts went right back to winning after one bad season.