The answer to the problem is to actively pursue “universalism” — avoid programs that target poorer Americans or that have eligibility cutoffs for people with too much income (“means testing” is the technical term) and instead deliberately distribute government aid across as broad a swath of the population as possible.
A universal basic income — which would give the same monthly cash benefit to every man, woman, and child in the country — is the extreme version of this idea. But universalism is an approach, not a policy, and can be pursued to varying degrees. For instance, one could imagine a program that gives the same monthly cash benefit to every family with a child, per child, regardless of income. You could extend that framework to the rest of the welfare state as well, with similar programs for every student, every retired person, every disabled person, and every person caring for a disabled or sick person. Single-payer systems are an example of universalism in health care.
The closest America has come to universal programs are things like Social Security and Medicare, which go to everyone over a certain age. And while resentments of the poor in general and African Americans in particular have certainly been used to prevent their expansion (Social Security excluded 65 percent of the black workforce and only 27 percent of the white workforce when it was implemented in 1935, and the gaps weren’t closed for almost two decades) there’s little evidence from American history that they can fracture universal programs once they’re in place. Social Security and Medicare, much to the chagrin of many reformers, are pretty much untouchable in U.S. politics. Meanwhile, politicians have cut AFDC and other targeted income supports in the last few decades, sometimes severely.