The U.S. political parties have a troubled relationship with "cosmopolitanism" as such an idea is not embraced within the U.S., as such. There are, however, elements of American political thought which are related to cosmopolitanism, and which carry over.
Pro-Cosmopolitan Trends in the U.S.
The idea of a global liberalism is codified in U.S. charters, like the the Declaration of Independence, which begins its second paragraph with:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
Liberal globalism was further restated by Martin Luther King, Jr. when he said:
Injustice anywhere is a threat to justice everywhere.
More recently George W. Bush asserted:
Are millions of men and women and children condemned by history or culture to live in despotism? Are they alone never to know freedom, and never even to have a choice in the matter? I, for one, do not believe it. I believe every person has the ability and the right to be free.
Elements of this idea are strong in U.S. political parties, but especially in the groups that are known as "Fighting Liberals" or "Neo-conservatives." Both of these groups grew out of the anti-Communist Left in the U.S. during WWII, and, as embodied by people like Paul Wolfowitz(R) and Anne-Marie Slaughter(D), are the dominant element of Foreign Policy in both parties
As noted in the question itself, the U.S. has been a major proponent of the development of many of the international institutions, including but not limited to the United Nations, NATO, the IMF, the WTO (which grew out of the GATT) and the World Bank. Many people--in the U.S. foreign policy community especially--believe that institutions are a good way to restrain revisionist powers (states that will wage war to change the global status quo) and as such the U.S. has pushed for the formation of these organizations and is a key player in all of them.
In general the Democratic Party is home to more people who believe in global institutionalism. Part of the reason for that is that "fighting liberals" are less willing to use force to impose liberalism abroad than "neo-conservatives" and international institutions are a constraint on that power.
The best illustration of this reaction were the priorities in the lead up to the invasion of Iraq in 2003 or the bombing of Libya in 2011. The Republicans were primarily concerned with passing authorization of force in the U.S. in both cases, and in the case of Iraq were totally willing to disregard international institutions. In Libya, the Democrats were by in large willing to bypass domestic institutions, so long as international support was received.
International Free Trade
International free-traders are not cosmopolitan in the sense that is normally meant, but are cosmopolitan in prescription. Free-traders treat all goods as essentially equal, at least as far as their country of origin are concerned, and have worked to reduce barrier to trade, and in some cases international movement.
Both political parties have a history of protectionism--Democrats protecting labor, and Republicans protecting agriculture--but for the most part both parties are free-trade advocates now. That may be changing however, as some Republicans have led opposition to the TPP (a free trade organization) and both Hillary Clinton and Bernie Sanders have expressed opposition to the agreement, even though Hillary Clinton helped to negotiate it.
Closely related to Liberal Globalism in outlook, but at great variance in prescription, American Libertarianism believes in free trade, and the innate right of all human beings to seek what is best for them as individuals. Libertarianism has its roots in the rugged individualism of the American frontier--perhaps best personified by "Teddy" Roosevelt--and in the Yankee can-do attitude that characterized New England in the 19th century. Libertarianism differs from Liberal Globalism in that it is utterly unwilling to do anything to export its values abroad other than proselyting. Libertarians also advocate for open borders, but are usually somewhat hostile to international organizations.
Libertarianism is small but present in both political parties. Usually during each political cycle at least one candidate in each party emerges to carry the Libertarian banner, and one or more will defect to the Libertarian party. This year, Rand Paul--and in the past his father, Ron Paul--is the highest profile version of this. Bernie Sanders has, in the past, run with elements of Libertarianism in his campaigns, especially when he was mayor of Burlington, but he has largely eschewed such policy now. For the time being, Libertarians seem to be more common amongst Republicans than Democrats, but not by much.
Not to be confused with the Universal church, which is now part of the Unitarian-Universal church, Christian Universalism is the idea that all humanity are Gods children and need salvation. This manifests itself in the widespread missionary efforts that many U.S. denominations engage in. Rather than participating in colonizing Africa or China the U.S. largely sent numerous missionaries. The "missionary generation" in China served as a formative tool of Sino-American relations in the early 20th century. Threats to missionaries ultimately served to involve U.S. troops in the Boxer Rebellion as well.
Christian universalism is present in both parties, but it is much stronger in the Republican Party. Not all religious groups are proselytic or universalist in outlook, though. One group that breaks overwhelmingly democratic are african-american Christian denominations (e.g. A.M.E.).
Anti-Cosmopolitan Trends in the U.S.
While the origins of American Exceptionalism can be debated, it is more than just outsized patriotism. Under American exceptionalism, the U.S. has a special mission, either to serve as an example to the rest of the world, or to provide services no other country can provide. Modern American-exceptionalism can be summarizes by two quotes, one from each political party.
If we have to use force, it is because we are America. We are the indispensable nation. We stand tall. We see further into the future.
The past few days when I've been at that window upstairs, I've thought a bit of the 'shining city upon a hill.' ... That's how I saw [the United States], and see it still.
And how stands the city on this winter night? More prosperous, more secure, and happier than it was eight years ago. But more than that: After 200 years, two centuries, she still stands strong and true on the granite ridge, and her glow has held steady no matter what storm. And she's still a beacon, still a magnet for all who must have freedom, for all the pilgrims from all the lost places who are hurtling through the darkness, toward home.
At times, this viewpoint has led to cosmopolitan tendencies, such as establishing international organizations, or participating in international aid because of a "unique" capacity, but at others, leads to the U.S. to isolate itself, since it has a responsibility which it must shoulder, alone if necessary.
American exceptionalism is overwhelmingly dominant in both parties. That view is tempering over time, however, and surprisingly the changes are fastest amongst Republicans. Nevertheless, the strength of the idea is illustrated in how Barack Obama had to back away from even the implication that he did not believe in American exceptionalism. Another example of this is the American high courts "allergy" to citing foreign court rulings on issues of constitutional law, which transcends party boundary. At the same time, justices are completely willing to cite amicus curiae from ordinary Americans.
American nationalism can, but does not necessarily, take the form of American exceptionalism. As a non-ethnically defined nation, American nationalists can come in all shapes, sizes and ethnic backgrounds. The idea of an American nation, as with all nationalism, is opposed to cosmopolitanism.
This is [the overwhelmingly dominant view in the U.S.] Although it is stronger in the Republican Party than the Democratic Party, Americans view themselves as distinct.
The basic idea of Cosmopolitanism--that we are all holders of similar ideas--is completely anti-thetical to Conservative Christian ideology. Muslims, Jews, Buddhists, Zoroastrians, Atheist all have ideas that are different from Christianity in fundamental ways, just to name a few. Even so-called 'liberal' or 'progressive' Christianity can get riled when people contend that Christian values are the same across the board.
This is by far more influential in the Republican Party. The exception, as above, is African-American churches in the Democratic party.
As with all countries, parochial interests veto cosmopolitanism. Historically, labor was a major obstacle to globalization, but many other groups are becoming more parochially inclined.
There is almost no support for Cosmopolitanism in the U.S. in either party. American Exceptionalism and American Nationalism alone are so widespread that cosmopolitanism gets drowned out. The only places where you here discussion of such ideas is in "transnationalist" enclaves, such as the ultra-elite universities, and the super-wealthy. Democrats dominate in academic environments, and while there are super-wealthy Republicans, some other principle such as Christian Conservatism trumps any cosmopolitan tendencies. As such there is only very tepid support for cosmopolitanism as such in the Democratic party. Note that none of the candidates for president in the Democratic Party are even tipping a hat towards cosmopolitanism.
That said, both parties have incentives and ideologies which can be amenable to the cosmopolitan ideal. The globalist viewpoint is not limited to cosmopolitanism as an idea, and all of the recent U.S. presidents, and most of the current candidates have at least some policies consistent with globalization, globalism and cosmopolitanism.