It would seem noncontroversial that you’d want to exclude people from immigrating who will have to be supported by existing citizens, but even that proposition is now politically incorrect. From Joe Guzzardi at progressivesforimmigrationreform.org:
No sooner had the Trump administration announced its intention to impose a new rule that will deny public benefits like food stamps and Medicaid to some legal immigrants, then 13 lawsuit trigger-happy states filed action against the Department of Homeland Security. Immigrants who are likely to rely on welfare benefits for their survival are generally referred to as public charges.
In their complaints, Colorado, Delaware, Illinois, Maryland, Massachusetts, Michigan, Minnesota, Nevada, New Jersey, New Mexico, Rhode Island, Virginia and Washington agreed in principle that the new rule would reverse a decades-old policy of granting affirmative benefits to immigrants, would violate the Immigration and Nationality Act, and is “arbitrary, capricious, and an abuse of discretion.”
New York Attorney General Letitia James, who said her office also would sue, struck an emotional note when she said that eliminating benefits to new permanent residents is “un-American, anti-immigrant, and unlawful…”
But contrary to critics’ high-pitched objections, according to Section 212(a)(4) of the INA, any individual seeking admission to the United States, or applying for a Green Card, is ineligible if s/he is likely to become a “public charge.” But, the guidelines have been ignored for decades and throughout Republican and Democratic administrations. Public charge first appeared on the scene in 1645, when Massachusetts was still a British colony, and European governments paid for their poorest residents to migrate to America. The legislation reflected colonists’ belief that residents were obligated to care for themselves.