Too late, at just one single hospital, Dallas Parkland, there are over 900+ babies from illegal aliens being delivered every single month, as Texans are forced to pay the bills. That's about 12,000 anchor babies per year, at just one hospital in Texas.
While you weren't looking, it's already happened.
Well, we can easily change the immigration law to prohibit those who were in the country illegally and gave birth to an anchor baby to be forever banned from any sponsorship or coming to the US.
The problem at Parkland is that my county funds a public hospital via the property tax. This depresses both property values, as well as puts a strain on the system when, as you pointed out, illegals (both from Dallas County and neighboring Collin and Denton counties) come to get free care.