In focusing strictly on the employment effects, however, Rodrik elides the biggest gain from trade -- lower prices.
Since Drezner’s point reflects a common misunderstanding about the effects of trade, it is worth some explication.
Advocates of globalization love to argue that free trade lowers prices, and the argument seems sensible enough. Think of all the cheap goods from China that we can buy at Wal-Mart. But anyone who understands comparative advantage knows that free trade affects relative prices, not the price level (the latter being the province of macro and monetary factors). When a country opens up to trade (or liberalizes its trade), it is the relative price of imports that comes down; by necessity, the relative prices of its exports must go up! Consumers are better off to the extent that their consumption basket is weighted towards importables, but we cannot always rely on this to be the case.
Consider your typical Argentinian for example, who consumes a lot of wheat and beef. Since these are export products for Argentina, free trade implies a rise in the relative price of the Argentine consumption basket. (The gains from trade are still there, of course, but they derive from the usual allocative efficiency improvements, not from lower prices across the board.) And in the U.S., the Wal-Mart effect has to be qualified to take into account the fact that the relative price of the goods that the U.S. exports (including for example agricultural commodities) is higher than it would have been absent trade. Similarly, when the U.S. gets better market access abroad for its agricultural exports (a key demand under the Doha round), you can be sure that this will raise domestic prices for these goods, not lower them.
Of course, if you are running a huge trade deficit like the U.S., you can have cheaper prices all around—for all to go on a consumption binge as long as the party lasts. But this is hardly the argument we make when we teach the benefits of free trade.
Postscript: It is interesting that many of the commentators below have appealed to scale economies as an explanation of how production costs and prices can fall all around as market size expands thanks to trade. Yes, this is a possibility. But scale economies raise a whole set of new conundrums (which is why I had stuck with the standard comparative advantage story). In particular, since scale economies are not compatible with perfect competition, we find ourselves in a second-best world with all kinds of strange possibilities. Opening up to trade can leave some countries worse off, and in general trade-distorting policies like tariffs and subsidies can make individual countries better off. So be careful how you describe the world we live in...