I think the Doorway is reason enough, really. I mean, it's a gate to hell and the Amazons are supposed to protect it. Hard to get better reasoning than that; if that thing blows open it's an extinction level threat.
And depending on the continuity there's other creatures on the island; harpies, cyclops, hydras, etc., and they all consider the Amazons either rivals or food or both. So there's plenty of reason for the warrior culture there too. Can't have the Amazon population decline because they keep getting eaten.
Personally I have little interest in Themyscria being treated as a utopia anyway. They should be more advanced than we are in many, many ways, both sociologically and technologically. But they should not be a perfect society that has moved beyond *all* the concerns and flaws we have. I *want* my Amazons to be a little hypocritical, a little bloodthirsty, etc. Gives them room to grow, and that makes for a better story than them just being perfect and flawless from the start. I think Diana's mission has a lot more weight behind it when the cultural exchange goes both ways. Yeah, Man's World has a lot to learn from the Amazons, probably more than they have to learn from us. That shouldn't mean they have *nothing* to learn from humanity though, or flaws of their own to overcome.