The first thing is to realize that there are probably always going to be Islamic extremists, just like there are always going to be people like Timothy McVeigh.
Violent fundamentalism is a reaction to the modern, secular culture. This culture seeks to impose control on as many people as possible through economic, cultural, and (sadly, often) military means. The culture of this world is forward-thinking—it needs to be, because that fuels technological progression, which in turn fuels economic development in a self-sustaining cycle. You need to step back a little to realize how radically different this is from the ideology of fundamentalist Islam, which basically sees the apex of human culture and history as frozen during the time of Muhammad and the rightly-guided caliphs.
McVeigh, like Islamic fundamentalists, see modern, secular culture as a holistic, unified system. And they are absolutely right to do this. You can’t separate the technology, the economic structure, or the social ethics of the modern world from each other. They all reinforce each other. “Moderate” religious people may claim to pick and choose from modern secular culture, but in reality they are just buying into the entire system wholesale, perhaps while keeping a bit of their traditional religiosity as “flavor.” (Look at how little American Christians base their worldview on what the Bible says. Also look at “moderate” Muslims who drink booze and advocate religious tolerance.)
This doubtless horrifies people like Osama bin Laden, who—rightly—sees modern culture as an eroding influence on traditional, fundamentalist Islam. An Enlightenment is the last thing such people want. An Enlightenment is buying into the system.
I don’t really know there’s much else to do but to continue what we are doing—that is, continuing to improve and progress our own culture. There will always be barbarians at the gates. But we can make our culture so advanced that more and more of these barbarians will want to buy into it. Dubai is a failed experiment in this regard—we exported the worst elements of capitalism to the city and now it’s probably going to end up a wasteland husk—but it was instructive, because at its height, Dubai really was a counterbalance to Islamic extremism. People wanted to live there instead of in Taliban-ruled Afghanistan. What we shouldn’t do is seek to impose our culture militarily, as that seems to have the opposite effect that we want.