When and under which president?
I’m not sure that you’re asking about the student aid system, which is how the other respondent interpreted your question. Most of the questions in this area of the site are related to finding money to pay for higher education, so that would explain the assumption.
If you’re asking about the welfare system in general – the social program that provides cash or other assistance to the poor, then you need to focus your attention on the administrations of Herbert Hoover and Franklin Roosevelt. "Welfare", as we know it today, was introduced in the United States by Hoover, but became a cornerstone of FDR’s expansive New Deal program.
The United States of the early 1930s suffered unemployment rates as high as 25%. More than 5000 banks collapsed, robbing the savings of millions, and the operating capital of thousands of businesses. Schools all over the US shut down, as the cities and states weren’t able to cover the costs of facilities and teacher salaries.
Hoover first attempted to address the problem by applying the same kind of "trickle down" approach that we see heralded today – he created the Reconstruction Finance Corporation, which injected over $800 million in the banks, the railroads, the insurance companies and farm mortgage entities. Despite the generous lending, the RFC failed to simulate anywhere near as many jobs as the administration had anticipated, and public protests and riots began to sweep the country.
Much as today, there were highly charged debates about the advisability and efficacy of using public money to provide relief from unemployment. There is some controversy as to whether Hoover, himself, exclusively promoted the ideas of private relief and volunteerism, rather than payments from the public dole, but there is no question that these payments began at the tail end of the Hoover administration, and grew exponentially as a keystone part of the Roosevelt New Deal.
I hope that helped you get started with your research. Good luck to you.