I commonly hear (though, interestingly, not from the staff at the AFL-CIO) that "globalization" is a key factor in understanding the decline of American labor unions. Erik Loomis makes the case at some length.
I'm quite skeptical. If you compare the United States to other rich countries what stands out is that we engage in relatively little international trade. Not coincidentally, we're much bigger than most other developed countries. Los Angeles exports media and tourism but imports cars and airplanes. Michigan exports cars and cherries but imports airplanes. Washington State exports airplanes and software but imports cars and media. So a country like Finland with a smaller population than LA County or Michigan is obviously going to need to import tons of stuff. In exchange they have a few major exports. And yet you see that Nordic countries have both very high levels of unionization and a lot of openness to trade. Canada isn't a super-unionized country but it is both more unionized than the United States and more exposed to trade.
What's probably true is that not globalization but policy that has the United States running a perpetual trade deficit has been bad for labor unions by depressing manufacturing employment. That speaks not to the level of trade, but to the structure of the American tax code and to the nature of American currency policy.