Not it doesn't really.
Dollars are in fact just pieces of paper. It doesn't cost a dollar to print a dollar, which would have to be true for the "broken window fallacy" to be applicable.
Just be goddamn rational. If Mexicans get dollars, they put them in the bank, converting them to pesos. Those dollars are then provided to Mexican businesses to purchase goods from America. It's what American dollars are for.
Those dollars might float around for a little while, but they're for buying American things. And if some Mexican gets dollars and just fills a bathtub with dollars. Well, he did a pile of work, and got a pile of pieces of paper for it. It's his loss if he doesn't spend it, not America's. And if you think America did lose out because some Mexican is hoarding 1 million pieces of paper from America and won't use them to buy things, well then the US government could just as easily print 1 million new bits of paper and buy things with that, and give them to American, or Mexicans for that matter. The economic effects would be identical to if someone else bought the same things with the same amount of money.
And if a dollar gets "lost", all the nation has actually lost is that bit of paper, nothing more. The cost of printing money is near zero relative to their face value. The idea that money is a tangible thing in the sense that a physical commodity like gold is, where you can do zero-sum logic like "if Mexico gets our gold, we have less gold". Well that logic doesn't work for a fiat currency. You can't make gold for less than the market price for gold. The government can in fact print money for less than the market price for money.