I think one of the various Doctor Who versions would be a good candidate.
My guess would be the Jon Pertwee version.
In T2, he did not save the world as evidenced by the fact that there was a T3. And in T3 all he did was save two people from nuclear war.
Teri Garr and Jeffrey Jones get one point.
No mention of Kim Possible or the Powerpuff Girls? They save the world (and the day) more times before breakfast than Bond or those other guys did in their entire careers.
You know, probably the best way to go about definitively answering this would be to make a list of movies and TV shows in which the world is saved, and which character does the saving.
That said… howzabout Sean Connery? As James Bond; As Ramirez in Highlander 2; and… uh… Allen Quatermain… Hmmm… That’s three, at least.
On the other hand, Bruce Willis - not in Die Hard, but the other three mentioned, plus arguably Hudson Hawk - wherein he prevents the destruction of the global economy.
I’m gonna go with Fox Mulder and Dana Scully…
As Austin Powers and who else? I mean, the world wasn’t exactly in danger in Wayne’s World or So I Married an Axe Murderer.
Personally, I think it’s probably Ahnold, though some of his saves are a bit debateable. But I would count T2: He did stop a nuclear war. Just because the world didn’t stay saved, doesn’t mean that it wasn’t saved for a while.
Perhaps they were baker’s monkeys…
And in Die Hard 1, 2, and Vengeance, he saved a building, an airport, and various portions of a city. But he’s working his way up. In the next Die Hard, he ought to be saving at least one U.S. state. Or maybe another country?
Will Smith has at least 3 saves – Independence Day, I, Robot, and Men in Black.
And Kevin Bacon probably hasn’t saved the world the most times, but I’m sure he’s within a half-dozen degrees of being the winner. 
There are three Austin Powers movies.
Ben Affleck’s got two saves; Armageddon and Clear and Present Danger. And a half point for defeating the Nazis and the Japanese almost single-handedly in Pearl Harbor. Minus a point, however, for trying to destroy the world in Dogma.
Hm. How would one count points for Mel Blanc, the Voice of Looney Toones? 
So is Arnie the winner then, followed by Will Smith?
And I’ll ignore that slur about poor Frodo. Yeah, he saved the world. Yeah, it was kind of an accident. You could say the same for most heros.
It seems almost nobody has understood the rules stipulated in the OP.