Writing at Foreign Policy, economist Daniel Altman does a good job explaining the benefits of free trade:

basic economics teaches that two people who trade with each other always end up better off. Why else would they trade in the first place? ... the idea is that there are gains from trade to both sides whenever a transaction occurs. Realizing those gains by buying and selling goods, services, assets, and labor is one of the keys to economic growth.

However, he then expresses some concerns with the impact of free trade.   He says it does not affect everyone equally:

Though two countries that trade with each other will also achieve gains from trade, their people won't necessarily share those gains equally. Indeed, both countries will be better off overall, but inside each country there will be winners and losers

In his view, we should follow a policy of free trade, and, in fact, we could do so if only we would compensate the losers:

But free trade needn't be such a divisive issue. At the national level, the benefits from free trade always outweigh the losses; this has to be true, since each new transaction creates its own gains from trade. Put another way, a country that opens its markets is always richer as whole. And so here's the genius part: It should be possible for the winners to compensate the losers so that everyone is better off, or at least no worse off than they were before. ... [This] is an essential part of the process of opening markets. Done right, it practically guarantees that everyone in a country can and will support free trade.

We already do compensate the losers, of course, but he says we don't compensate them enough:

Sure, there are programs such as Trade Adjustment Assistance (TAA) in the United States and the Globalization Adjustment Fund in the European Union. But they're tiny and relatively ineffective.

He proposes the following:

One idea would be to identify the prospective winners of a new trade agreement and ask them to contribute lump sums to a fund that would compensate the losers. The trade agreement would go forward much as buyouts do in the stock market; if enough of the winners signed up and contributed, the rest of them would be compelled to pay, perhaps on their annual tax bills. Then the fund -- likely to hold a lot more than $1 billion for any major agreement -- would be divvied up between the losers.

I appreciate his efforts to get the public on board with free trade. It's always nice to find allies in this fight!  But I have doubts that his approach is a good one.  (I'm going to put aside any general issues related to compensating people who have lost jobs or otherwise suffered from a bad economy.  I'm not really qualified to weigh in on this.  Let me just focus instead on the trade related issues.)

First off, I'm skeptical that this plan can be administered in a satisfatory way.  A government agency is supposed to sift through all the claims that individuals have "won" or "lost" from a trade agreement?  Perhaps the simple "won" vs. "lost" issue could be determined, although it would be hard.   But what about the degree of "winning" or "losing"?  How much compensation is due?   That will be challenging, to say the least.

Second, we already have a number of social programs designed to help people who are in a difficult economic situation.  Why do we need additional programs for people harmed by trade?  What is special about trade that merits extra government support?  I don't see how it is worse to have lost your job due to increased trade flows than due to changes in technology, and why anything additional is merited.  Extra money for trade losses seems unfair to those who have lost jobs for other reasons.

And finally, I think it is important to distinguish two kinds of trade harm.  You might have an industry that never experienced foreign competition, due to natural barriers, such as distance.  If new technology (the Internet, shipping) makes trade possible, the harm that may result could be thought of as unexpected.  By contrast, you might also -- and often do -- have an industry that kept its foreign competition at bay for years by lobbying the government for tariff protection or subsidies.  Through a trade agreement, that protection is removed, and the company faced with foreign competition suddenly does a lot worse.  In this situation, is there really any need for compensation?  These companies have been receiving special favors for years.  It's not clear to me why they need to be compensated when the government removes those favors.