I think it better that in times like these
A poet's mouth be silent, for in truth
We have no gift to set a statesman right;
He has had enough of meddling who can please
A young girl in the indolence of her youth,
Or an old man upon a winter's night.
Yeats wrote his in 1915, after being continually asked to write a poem about the war. Obviously tired, and starting to feel the deep futility that would gradually set in all across the European art world, Yeats offered this. It's little more than a shrug, a poetic "what's the point?"
I think history has proven Yeats a little wrong, and a little right. He's correct to say that poetry seldom influences policymakers, and does not deter them from the great folly of war. But that is not sufficient reason for a poet's mouth to be silent, as he must have felt (why else would he write this, and Easter 1916?). The poet-soldiers of the Great War captured for us all the moods around Europe at the time. The hope, the anger, some jingoistic schlock, the horror, the despondency, everything one could feel from war. Poetry's purpose is not to shape state policy.