It's a complex problem. Many of the illegal aliens do pay taxes using bogus SSNs. Downside is that if/when they file, their income level is normally so low that they qualify for EIC and get more than they pay into the system. But, the results are the same as if they were here legally and did the same job/same pay.
If we truly had border security, illegal aliens wouldn't be such a polarizing issue.
My wife works in a different state. We get the joy of filing two state tax returns and having a portion of her income result in taxes for two states. Would be great if her career field was one that the state exempted.