This person has a child.
30k in New York city is not very much. I think it would be/will be hard to get by, but might not be impossible. You would probably not be the 1st grad student with a child to do this. Health insurance is important to have if you have a child...you should see if you would be able to get Medicaid, especially for the kiddo. If you can get that and other assistance like food stamps, etc. you might be able to make it, but it's going to be hard.
Also, if you are working in Manhattan could you live somewhere else like New Jersey or Brooklyn and commute into Manhattan?
Also I wonder if your wife could take in other kids to watch or do something else to get a little bit of extra money.
Also I wonder if there are other options of places to do the Ph.D. I just say this because NYC is probably the most expensive place you could possibly live in the U.S.
Is there someone at the school/place you got offered the Ph.D. who helps people deal with issues like the finances, finding affordable housing, etc. Also, does the university have any subsidized housing for families that you could live in?