I'm sure your laws are different but here's what I did in 93 to sponsor a former-Yugoslavian family. They were actually war refugees so it's different unless the US classifies the state of South Africa a war zone. Anyhow, we needed $10,000.00 in the bank (in cash not in equities), a place for them to land (we got them a cheap apartment), and a promise to look after them until they got on their feet (hence the 10K so they wouldn't be on welfare).

Since they were refugees, they were allowed to go on welfare after 6 months. They both did, for about 2 months for the father (I got him a job gardening and he went off welfare), and a few years for the mother who was raising an at the time 5 year old who was shell shocked.

Both parents went on to get their PHD's, one in Sanskrit and it a Professor of Sanskrit now, and the other in Hebrew. So, he's a prof and she's a really high end yacht detailer (not much calling for profs for hebrew in BC). They both became (after 10 years) Canadian citizens, both have other children (their relationship fell apart...meh) and are happy, well adjusted Canucks now.

I'm sure the process can not be that much different in the US.