The 1993 WTC bombing alone didn't really change US foreign policy all that much. We get a much more accurate representation of the bombing's significance when we view it as a single instance in a larger pattern, so please forgive me for expanding your question's scope. If we look at the WTC bombing together with other acts of terrorism throughout the 1990s we can see that terrorism came to hold a great deal of influence in regard to how Americans viewed the nature of global power and the United States' role in the world during the post-Cold War era. (As a sidenote, this is paraphrased from my Master's thesis.)

First, let's put terrorism in the 1990s within the context of how Americans viewed the post-Cold War era. With the collapse of the Soviet Union, the international system appeared to fragment as weaker nation-states and non-state actors all gained relative influence within the post-Cold War order. Instead of the apparent simplicity and stability of the bipolar Cold War order, Americans were thrust into a much more complex international system and were consequently left without a clear understanding of the United States’ role in the post-Cold War world. Foreign policy analyst Richard Haas said of the post-Cold War era that “A sure sign that experts are encountering difficulty with figuring something out is their use of ‘post-ʼ as a prefix. Such a label reveals that people know only where they have been, not where they are now, much less where they are heading.”

Foreign policy intellectuals inside and outside the government offered various interpretations about the state of contemporary global affairs in the post-Cold War era. In general, these views can be lumped into two camps. One view of world affairs that emerged out of the collapse of the Soviet Union throughout 1989 and 1991 was a triumphalist interpretation. Its proponents believed that the primacy of the United States and Western liberalism became solidified after the collapse of communism. Prominent intellectuals in support of this view included the political scientist Francis Fukuyama and Washington Post columnist Charles Krauthammer. In contrast, some foreign policy analysts viewed the post-Cold War order as being defined by a clash between the unmodern and the modern—in the words of political scientist Benjamin Barber, a conflict between “Jihad vs. Mcworld.” Most prominent among this intellectual circle were Barber, New York Times columnist Thomas Friedman, political theorist Samuel Huntington, and foreign policymakers within the Clinton administration. Within each of these interpretations, foreign policy elites sometimes drew vastly different conclusions about their significance for the United States and global politics. Whereas policymakers within the Clinton administration saw modernization as a force for democratization, Benjamin Barber saw it as something wholly indifferent to democratic values; and while Fukuyama professed the end of large scale conflict, Krauthammer believed the probability of war would increase during the post-Cold War era.

Throughout the Clinton era, repeated acts of international terrorism against American targets abroad and, for the first time, within the United States weakened the triumphalists’ narrative. In 1989, political theorist and former director of the State Department Policy Planning Staff, Francis Fukuyama, wrote an article in the policy journal National Interest titled “The End of History?” He argued that World War II and the Cold War saw the proliferation of American political and cultural ideologies throughout the world. The solidification of American political and cultural power in turn led to changes in ways that the West perceived the non-Western world. The Americanization of the so-called Third World led to a “great awakening and “reawakening” of non-Western cultures through U.S. modernization programs and the creation of intergovernmental organizations like the United Nations. Fukuyama argued that, through American modernization and the integration of Third World countries into the international order, non-Western cultures would ultimately transform into liberal democracies, defined by Fukuyama as being made up of free market economies, democracy, equality, the rule of law, a burgeoning civil society, and the prevalence of honorable behavior and humane values. The rise of liberal democracy represented to Fukuyama “the end of History,” the point at which humanity reached the apex of civilization and all major global conflicts would end. In his famous words, Fukuyama ultimately concluded that what the world was witnessing in 1989 “not just the end of the Cold War, or the passing of a particular period of postwar history, but the end of history as such: that is, the end point of mankind’s ideological evolution and the universalization of Western liberal democracy as the final form of human government.”

Terrorist attacks throughout the 1990s weakened such an optimistic interpretation of the post-Cold War era. In 1993, Islamic militants parked a truck filled with explosives into the parking garage of New York’s World Trade Center. When the bombs went off, the explosion ripped through six stories, leaving six dead and 1,042 wounded. Only a few months later, U.S. officials narrowly stopped Islamic militants from what would have been a catastrophic, highly-coordinated terrorist attack that involved the simultaneous bombing of the George Washington Bridge, United Nations headquarters, the Holland Tunnel, and the Queens Midtown Tunnel. In June 1996, nineteen American military personnel housed at Khobar Towers in Dhahran, Saudi Arabia, died in a suicide bombing. In 1998, al-Qaeda operatives simultaneously bombed the U.S. embassies in Nairobi, Kenya and Dar es Salaam, Tanzania, killing 213 people including twelve Americans, and wounding over four thousand. In the final months of Clinton’s administration, al-Qaeda operatives engaged in another act of terrorism against Americans abroad, piloting a bomb-filled boat into the USS Cole, a Navy destroyer harbored in Aden, Yemen, that killed 17 American servicemen.

The American foreign policy elite's response to these terrorist attacks indicated that Americans saw the post-Cold War global order as more, not less, prone to conflict. Furthermore, with the United States the sole remaining superpower, its dominant role within global affairs would make it a main target for politically marginalized states and non-state actors. Unconvinced by the view that liberalism had triumphed and that it would inevitably spread across the world, Americans throughout the 1990s searched for a more convincing view of the United States’ role in the post-Cold War global order. This purpose would be shaped primarily by the establishment of a new global threat to American values and interests that would replace the Soviet Union. As a Time article stated as late as June 2000, America was “In Search of an Evil Empire” that could act as a foil to the United States’ global values and interests. Shaped by the United States’ encounter with international terrorism in the 1980s and 1990s, America’s foreign policy elite came to view the post-Cold War global order as a conflict between the unmodern and modern worlds.

The Clinton administration framed its understanding of the post-Cold War global order within a modern/unmodern dichotomy that manifested into a grand strategy meant to succeed America’s Cold War containment policy. Above all, the Clinton administration believed that the spread of democracy and free markets would best promote American global interests in the post-Cold War world. Consequently, Clinton and his foreign policymakers crafted a foreign policy strategy founded on what they labeled “democracy enlargement,” which stressed the United States’ promotion of free market values and democracy throughout the world.

The Clinton administration believed that the United States should work to transform what it labeled "rogue states" through the global spread of liberal political, social, and economic institutions. As Anthony Lake argued, America’s dominant position within the international system meant that “the United States has a special responsibility for…transform[ing] these blacklash states into constructive members of the international community.” Modernizing the world’s “backlash states” through the encouragement of democracy and modernization would, in turn, weaken those states’ reliance on terrorism and consequently improve American national security.

Such a state-based approach to terrorism was not wholly applicable to the international terrorism of the post-Cold War era, which was much more often conducted free of any influence by rogue states. As such, many of America's foreign policy elite outside of the administration never really latched on to Clinton's understanding of terrorism. Losing its state-based narrative, Americans came to view international terrorism as part of what political theorist Samuel Huntington referred to as a "clash of civilizations." In particular, many of the American foreign policy elite outside of the administration came to view terrorism as inherently tied to Islam. (If you would like sources, just ask). It's from this section of the foreign policy elite that existed outside of the Clinton administration that is perhaps most historically relevant, since it provided a foundation from which Americans would come to make sense of the attacks on September 11th. Ultimately, it's from the American encounter with international terrorism throughout the 1990s that the Bush administration was so effectively able to construct a new American grand strategy centered around the threat of Islamic terrorism.

I really cut down on the last section since I wanted to get it all in one post. If you would like it to be more in depth, I certainly can post more.