We could stop this nonesense immediately if we wanted to. If the Secretary of the Treasury issued an order prohibiting their use as identification for banking transactions involving US banks, they would be of no use to illegal immigrants. Last I checked the federal government has the power to regulate the money supply and banking. Without banks, illegal aliens don't have a secure way to send money back home.
It would also help if the state required a social security card to get a driver's license.