If my husband and I co-sponsor my son's wife, what are our obligations for the future?

If the immigrant receives any “means-tested public benefits,” you are responsible for repaying the cost of those benefits to the agency that provided them. If the debt is not repaid, the agency can sue you in court to receive the money owed. When in doubt, ask the benefit provider if the benefit is a “means-tested public benefit.”Also and more importantly is the J visa she's on subject to the two year home requirement. If so then that could pose problems if she marries then tries to adjust to her green card.