There are different classes of immigrants, and there should be different rules for each class, because they are in very different circumstances.
1) Illegal immigrants, i.e. people who have entered without passing the immigration control at a port of entry. These are deportable, and so far as I know, have never been eligible for federal benefits. (Their young children may have qualified for WIC and other food assistance programs.) They may have qualified for state, local and/or private benefits.
2) Refugees / asylum seekers. These will in most cases need aid, and in my opinion should be given whatever help they need to settle in and become productive residents, workers and taxpayers. To deny them aid, or deny them permanent resident status if they get aid is counterproductive.
3) People that come here to marry US citizens – they should become eligible to aid as they will become permanent residents, almost immediately.
Immigration is a big tangle of interacting difficult problems made worse by bungling politicians over decades. Almost every suggestion you hear that sounds simple is counterproductive.
One of these days, I will write a long blog post about what I think could be done to start making things better, but not this week.