Not as powerful as Mearsheimer and Walt might believe. Walter Russell Meade reviewed their book The Israel Lobby in the recent issue of Foreign Affairs, and I pretty much agree with everything he has to say.
To summarize, he lists three major problems with their thesis and its execution, two of which I'll list here:
1) They don't define the Lobby at all, and their amorphous, wide-ranging definition of the Lobby is so unbelievably broad that it would include any group in America that recognizes Israel's right to exist no matter what its view on Israeli politics, and amazingly the authors themselves!
2) They don't show how the Lobby affects American policy towards Israel. They show how Congressmen and Senators routinely affirm Israel's right to defend itself in great numbers, but they don't show how the Lobby "persuades" them to do so. They also don't show how these congressional resolutions relate to actual foreign policy, which is formulated and executed by the Executive Branch and not Congress.
I'll take Meade's word for it that the authors are not anti-Semites, although these were formerly respected academics, teaching at the highest levels of academia who just produced a highly biased polemic with little basis in truth.
Read the whole thing.