As a person who deals with this kind of thing for a living all the time, let me tell you that there is nothing you can do to stop it. Unless the law changes, it will only be a revolving door and even then we cannot win.
I probably get people with fake documents (Social Security Cards, Permanent Resident Cards, US Driver Licenses, Mexico Driver Licenses, Birth Certificates, Mexico Passports, US Passports, Mexico Voter ID Cards, Mexico Consular Matricula Cards, Honduran ID Cards, and fake DL's/ID's from nearly every Central American County) at least a few times a week.
Sadly, under TX Law, we cannot arrest for fake foreign "state" documents. It has to be a fake federal document (i.e. a fake Mexico passport). If there are fake US documents, all bets are off, but not for fake Mexico DL's.
Most times they are not lying about their identity, and merely want to prove citizenship/residency, or that they can drive, but occasionally you do get one who is lying about their name and DOB, usually to hide past criminal history or warrants. We take them in on that, then they get out, may or may not get deported, and the cycle continues.
Not long ago, I got one woman for a fake SS Card and Permanent Resident Card. She had come here in 2000 legally on a B1 Visa and during her 10 year legal stay, had pumped out 5 kids, all of whom had SS Cards which she had in an envelope in the car. She carried that envelope around with all their birth certificates, Lone Star (EBT) Cards, Health Care Cards, etc. After I told her she was under arrest, she immediately fell on the ground and refused to move, screaming about her 4 month old child in the car. The child was in a car seat, not buckled into it, and the car seat was not buckled to the car... yeah... she really cared about her kid alright. We allowed her to call her sister in law as we were doing the paperwork, and her sister in law told her not to worry, because Obama would not deport her since she had 5 US citizen kids, and that he would pass immigration reform to keep her here. Oh, and she was driving out in a completely trashed out (on the inside) Cadillac Escalade.
A common scheme, and one that is impossible to beat, is when people assume another's identity. Example: A US born child dies say around 2 or 3 years old. His family is here illegally and cannot afford a funeral. A family with an illegal child goes to them and says "We will pay for your child's funeral, and in return, all we need is your child's SSN, birth certificate, etc." Basically, they pay for the funeral expenses, and in return, they or their children assume someone elses identity. It is nearly impossible to ever get caught and I am telling you now, we (as in law enforcement) cannot fight that.