Should-Read: Three points not made by Paul Krugman:
"We" collect revenue from a tariff--in addition to its other effects, it's a way for "us" to make "them" pay somewhat for some of the things our government does.
Over and above this, the dollar appreciation resulting from a tariff means that our terms of trade improve: we get to buy more stuff for each of our exports.
Foreigners will retaliate.
If not for (3), (1) and (2) would together create a pretty strong optimal tariff argument for a tariff. But (3) blows that case up. With (3), a trade war improves the position of import-competing industries and scarce factors both at home and abroad. But with even a moderate amount of increasing returns in the mix it would be a rare import-competing industry whose stakeholders benefitted, and a rare scarce factor that wound up with higher real incomes:
Paul Krugman: Tariffs and the Trade Balance: "The standard story... the capital account is determined by international differences in savings and investment opportunities, with capital inflows to countries that offer good returns...
...The real exchange rate then adjusts to ensure that the trade balance offsets these desired capital flows. In this simple story, a tariff shouldn’t lead to a lower trade deficit, as long as capital still wants to come here; it will just lead to a stronger dollar.... But this story is a bit too simple.... Protectionism... reduces trade flows; this means that larger real exchange rate movements are necessary to accommodate swings in the capital account; and these exchange rate movements themselves reduce the return to international investment.... This is a subtler channel than the crude notion that foreigners are taking advantage by selling more to us than they buy, and tariffs will fix that...