There are many enemies in the universe so Star Fleet is important. To name a few there are The Borg, The Makers, The Cardassians, The Romulans, And even the Klingons who are federation members.

War-like cultures would still exist even if the Federation believes they themselves have moved beyond that type of thing.

Yes, but they're not enemies like Nazi Germany was an enemy, or like Al Qaeda was an enemy, or even like Bonnie and Clyde were enemies. They're enemies like a hornet's nest is an enemy. (If you're really cynical, they're enemies like the Protoss are an enemy, seen from the perspective of an American playing Starcraft. They exist to make your universe more fun.)

Think of how far ahead of the Klingon Empire the Federation is. They routinely have a single man with delusions of grandeur outwit the entire Klingon race and nobody thinks this is odd. We know that in the logic of the Trek universe if it's the entire Klingon High Command personally at the helm of 100 warbirds versus Wesley Crusher and his high school science project that Wesley is destined to win, right? You don't even need to see the universe from the outside to come to that conclusion. Ask a Ferengi to handicap that contest. He'd tell you "Rules of Acquisition #47: never, ever, ever bet against the humans. There's no profit in it. Ever."

Think of some power multiplier, X, for demonstrating the geometric mean between civilization's level of ability to effect their wishes on the cosmos. Multiplying the Aztecs by Xs gets you the colonial Spaniards. Multiplying the Spaniards by X gets you the US. Multiplying the US by X gets you the lowest starship captain ever depicted in the series. Multiplying them by X gets you to the Klingons. The Federation is still several powers of X beyond them.

This implies that the Federation could probably wipe out everybody else with a for loop, and probably consider that low-intensity conflict resolution. But that wouldn't be fun. Why hit the Borg with the Lance of Judgment when there are still so many stories to tell about how Picard proved that passion and ingenuity beat raw technological dominance? The Federation loves such stories as only a technological superpower can.

An alternate take compatible with patio11's irreverent theory: war-like cultures exist only in the sense that some cultures have different LARP preferences.

Supporting evidence: how did the Klingon Empire (who are not actually part of the Federation) make it through their nuclear age without blowing up their planet? How do they continue to keep pace technologically with other species? They steal a lot of technology, e.g. they never invented the warp drive themselves, but that would only get them so far and wouldn't keep them competitive with the Federation.

Patio11's oddball theory of Star Trek as a documentary about a future LARP may not be correct, but it explains away these kinds of questions better than any "real" theory that I've seen!

Also I thought the Klingons were part of the Federation at one point during DS9 era? Or perhaps that was just an alliance...

There was indeed a military alliance between the Federation and Klingon Empire against the Dominion. A friend of mine (uncharitably) referred to the war against the Dominion arc as the writers, "exploring what Star Trek could be... if it was more like Babylon 5."