G.I. Joe: A Real American Hero is easily one of the most recognizable cartoon series from the 80's. For those of you who have been living under a rock, here is the basic concept of G.I. Joe: A counter-terrorist unit consisting of a number of members with various combat specialties are fighting an evil terrorist organization known simply as Cobra whose goal is total world domination.
While some might view the concept as goofy, I think the basic set-up for G.I. Joe could make for an interesting campaign. The player characters would be members of a counter-terrorist organization with each of them being a specialist in a certain form of combat. This group, who is funded by a mysterious entity, would be sent on numerous missions to stop an evil organization that wishes to take over the world.
I think the best way to handle this concept is to embrace the silliness of it. Don't shy away from the goofiness and use it to make the game more fun. You could have the evil organization kidnapping world politicians with plans to clone them and place those duplicates in seats of power so they can further their goal of world domination. You could have this organization create a machine that controls the weather and have them create massive storms around the world, hoping to destroy major military bases to weaken the major powers of the world. While you can have some serious elements in there and have drama between the character, I wouldn't shy away from the overt silliness.
For systems, I would suggest two: Savage Worlds and Spycraft. The reason I suggest these two systems is because both have books that allow you to already run these kinds of games (Strike Force 7 for Savage Worlds and Real American H.E.R.O. for Spycraft). If you want a more rules light system that emulates a more pulpy style of play, I'd suggest going with Savage Worlds. If you want some more crunch and espionage elements in your game, I suggest going with Spycraft. Both systems are a good choice and it just really depends on your preferred style of game.
What other cartoons do you think would make for a good framework for a campaign?