My point is that you make it more difficult, not stop it completely. If you squeeze the employers to demand the card and squeeze the money transfers, you squeeze the illegal immigrants. The ID card is a tool to take away their incentive. It's not the only thing, certainly.
Honestly, requiring an I.D. to ship money to Mexico is nothing. All they do is get any intermediary to do it. No squeeze at all.
As for the I.D. cards, everyone applying for a job should be required to present either a high-tech social security card (for citizens), or a guest worker I.D. (for immigrants). Immigrants will be required to carry that on them at all times, citizens won't.