Are there any rules (official or otherwise) that would govern trying to string together multiple games of Arkham Horror into some sort of a campaign structure? The idea would be for the investigators to retain some of their capabilities from game to game, and the game itself would have to ramp up in difficulty in some manner to compensate.
This may not be exactly what you're looking for, but check out the Scenarios FFG created for League play (here's a link to information about them on the Arkham Horror Wiki). They do ramp up in difficulty, and the Spanish league scenarios have a continuing story.
I'm sure you could easily adapt things so that your characters grow and develop through the games, perhaps by adjusting some of the stats or giving bonuses based on how the investigators "performed" in the previous scenario?
There is a Campaign mode on the board game geek website http://boardgamegeek.com/filepage/27501/arkham-horror-campaign-rules but as I read them I am trying to think of a way to set it so that the board don't get cleared between every round, to were the gates and the monsters stay on the board or something but when me and my friends tried it the second set you felt already doomed since we ended up starting with like 5 or 6 gates already open to where 1 or 2 would wake the ancient one right off the start not fun
You may want to check out Arkham Investigations on some alternate rules on creating "modules" for Arkham.
I think each module is self-contained, but you may want to pull some ideas from here on ramping up the difficulty.
It wouldn't really make sense to keep the gates on the board since they would be tied to the threat of an oncoming great old one. Presumably your characters would have had some downtime between one game in the campaign and the next. I imagine it would be like seasons in an ongoing tv-show. However, one could always make a roll for each gate to close some of the gates between games. Or giving each of the characters 1 chance to close each gate, going into it fully restored before starting the next game. This gives the opportunity to reduce the initial threat while making a fair deal of sense theme-wise.