I think we're ignoring a key point; althought this is a problem for the U.S., one of the big sources is Arab nations using anti-U.S. sentiments to distract attention away from their own corruption. It gives frustrated young men an outlet that is not perceived as a threat to the internal status quo, and helps the governments avoid detaining/killing/torturing these young men en masse (as they scream for the death of Zionism, crusaders etc, rather than for the death of their own prime minister, president, etc), which would only make internal tensions worse.

The countermove is fairly obvious, but wouldn't fly well in the media; diplomatic sanctions and loss of economic support to nations that spout anti-U.S./West venom from state-run media, and distribute free radios pretuned to the regional VOA and BBC freqs. And the carrot; diplomatic and economic considerations for nations that have free media or even-handed state run media. The real key to the carrot and the stick is making them rapidly adaptable, so changes in a nations actions (good or bad) don't go unrecognized.

Of course, the mainstream media would scream that this is interfering in freedom of press, a propaganda program, yada yada.