What connection is there between the New Deal and modern social policies?

Answer 1

The New Deal and the idea that it actually made things better during the Great Depression made it possible for the current social policies.
The New Deal had a number of Socialist thoughts and practices that this country had not before.
Now, the people think that its the standard duty of the government to do some of those things. The country worked fine before them, they only made the economy worse, and now its a standard fact.

The New Deal allowed the current Socialistic leaning of social policy in the US.

Answer 2

When the Stock Market crashed in 1929, then president Hoover did nothing, obeying the conventional wisdom that it was just a business cycle that should be ridden out. Proposals of government interference were labeled as "socialist" while the country slid into the Great Depression.
The New Deal created policies to employ vast numbers of workers and create a social safety net so the elderly and infirm wouldn't just be thrown out into the street when they could no longer work.
Today this social net has survived and expanded so citizens enjoy a higher standard of living. Government is largely expected to have a hand in guiding the economy and instituting policies so that disasters (natural and economic) don't devastate the nation like the Great Depression.