How is the money being sent now? I think if it’s thru banks that would be easy. Of course you wouldn’t be able to stop payments of cash, or if some third party is making the payments, but I think you could cut off a lot.
If the lines at my local Hy-Vee are any indication it's through grocery stores, check cashing stores, places like that which have Western Union outlets. Short of having everyone wiring money provide proof of citizenship or legal residence every time they want to make a transaction then I don't know how you would accomplish it. And can the government force private businesses to play police officer for them?