It's difficult to disentangle the government's response to the recession and the safety net itself, because they're both tied to contentious issues of government taxing and spending, and because the safety net is an important automatic cushion for the economy when it hits a recession.
But when Republicans came to power on Capitol Hill they seized on the fact that the economic crisis had caused deficits to soar, and used it as leverage to pick a generational fight over federal programs they dislike. Those programs cost money, they note, and by cutting or eliminating them we can reduce the deficits.
By making the deficits the issue, instead of attacking the programs on their merits, Republicans placed a deadly constraint on the government's ability to temporarily run deficits now in order to put people back to work. That's part of the reason why renewing the payroll tax cut was so difficult and why once-bipartisan initiatives like infrastructure spending -- initiatives that would help the economy enormously -- have become impossible.
Obama's big ask is that Republicans unlink the two, so that Congress can address the unemployment crisis, and in the meantime, the parties can fight over the appropriateness and sustainability of Medicare, Medicaid and other social programs. They won't oblige him, and the consequence is a persistently weak economy.