They dont do politics and war unless there's super villains, aliens, gods, or some other supernatural threat involved.

Is that war being spearheaded by Ares? They're on it

Is Grodd mentally controlling them to make the stupid ibferior humans kill eachother to make way for some sort of Ape Planet Rising? They're on it.

They may promote oeace talks, act as mediators, etc. But they arent going to fight another nations soldiers typically.

The exception: They fight crime. War Crimes are a crime. So technically, they'd get involve to defend civilians and fight against the war crimes and the like. So since you specifically mention genocide: yeah, they'd probably do something. But actually actively fighting the war is probably a no go.

Thing is, getting involved in stuff like this is complicated. This is often why they make up countries and have said countries be the ones to do bad things, typically while under the influence of a super villain.

Its also just not what the JLA is for, so naturally this rarely gets touched on, in or out of canon.