This is a complicated question that does not have a simple answer.
From a historical vantage point, the Democrats have pretty much always supported Free Trade, at least since the 1820s (when the modern party was founded under Andrew Jackson). The Jefferson administration embargoed European goods to prevent conflict during the Napoleonic Wars; this ultimately failed and the U.S. got pulled into its first quagmire, the War of 1812. The Jacksonian Democrats tended to be dominated by Southerners and immigrants prior to the Civil War, and typically wanted low tariffs to make it easier to import European goods. Whereas the Whig Party and early Republican Party typically wanted protectionist tariffs to help boost Northern industrialists.
After the Civil War, the Democrats generally still supported free trade. Although since the 1970s or so, this enthusiasm has definitely worn down due to the influence of organized labor and blue collar voters who often see free trade as a threat to their own livelihood.
The Republicans typically supported tariffs until after the Second World War, when a new spirit of internationalism (including both moderates and neoconservatives) took hold in the GOP.
Here is what the most recent party platforms say:
Democratic Party 2012:
President Obama and the Democratic Party know that America has the best workers and businesses in the world. If the playing field is
level, Americans will be able to compete against every other country
on Earth. Over the last four years, we have made historic progress
toward the goal of doubling our exports by 2015. We have taken steps
to open new markets to American products, while ensuring that other
countries play by the same rules. President Obama signed into law new
trade agreements with South Korea, Colombia, and Panama that will
support tens of thousands of private-sector jobs, but not before he
strengthened these agreements on behalf of American workers and
businesses. We remain committed to finding more markets for
American-made goods—including using the Trans-Pacific Partnership
between the United States and eight countries in the Asia-Pacific, one
of the most dynamic regions in the world—while ensuring that workers'
rights and environmental standards are upheld, and fighting against
unfair trade practices. We expanded and reformed assistance for
trade-affected workers, and we demanded renewal of that help alongside
new trade agreements.
Republican Party 2012:
International trade is crucial for our economy. It means more
American jobs, higher wages, and a better standard of living. Every $1
billion in additional U.S. exports means another 5,000 jobs here at
home. The Free Trade Agreements negotiated with friendly democracies
since President Reagan’s trailblazing pact with Israel in 1985
facilitated the creation of nearly ten million jobs supported by our
exports. That record makes all the more deplorable the current
Administration’s slowness in completing agreements begun by its
predecessor and its failure to pursue any new trade agreements with
This worldwide explosion of trade has had a downside, however, as some
governments have used a variety of unfair means to limit American
access to their markets while stealing our designs, patents, brands,
know-how, and technology-the “intellectual property” that drives
innovation. The chief offender is China, which has built up its
economy in part by piggybacking onto Western technological advances,
manipulates its currency to the disadvantage of American exporters,
excludes American products from government purchases, subsidizes
Chinese companies to give them a commercial advantage, and invents
regulations and standards designed to keep out foreign competition.
The current Administration’s way of dealing with all these violations
of world trade standards has been a virtual surrender.
As you can see, both party platforms express a mix of support for free trade with some reservations related to specific problems. For example, the Democrats are worried about unfair competition from low-wage countries, and the Republicans are worried about China.
This reflects, as an earlier response noted, that free trade is generally a Washington "Establishment" issue, that many ordinary Americans feel ambivalent about. Because both the Democrats and Republicans are both "big tent" parties, their platforms represent a compromise between "insiders" and "outsiders."