I have mixed feelings about it. My first reaction was to cringe, but later I thought it was pretty funny. As for whether they should have run it, I am undecided. I agree that it might have been more effective inside the magazine rather than the cover.
Part of my dissertation actually dealt with a really popular satirical poem that was completely misunderstood by most readers and that misunderstanding had dramatic political consequences. The poet's intention was to satirize racism, but the poem ended up being used straightforwardly, as support for racism. That's what this image reminds me of. It risks supporting the attitudes and opinions it's attempting to satirize. As journotraveler said, the New Yorker is usually more nuanced in its satire, but I think that the problem here is that, as ridiculous and over-the-top as the image seems, it's not at all far off from what a lot of people are thinking. It wouldn't be an exaggeration to those people; it would be a portrait!