How did the creation of public relief programs in the 1930s, such as the Social Security system, have a positive and/or negative effect on the social work profession?
In the time before the 1930's and the New Deal, social work was centered around efforts by private organizations and churches. Most people, at least those in government and who were wealthy, believed that the government should have no role in protecting or supporting any particular segment of society. These were the laissez-faire Presidents and sentiments of the 1920s "Charity begins at home" set.
By adding large scale social relief programs providing everything from food to jobs to retirement benefits, FDR gave social work permanency as a profession, and legitimacy in the eyes of a grateful public. It laid the foundation for social work to expand and continue and adapt to the changing needs of an immigrant population and those in poverty.
In terms of negative effects, one could also argue that when government took over the role of social welfare, it lessened peoples desire to donate to private charities. We can see a similar effect in European countries today with large social welfare programs, as many of the citizens feel that there is no need to donate to the poor as there are programs already available to help. This makes the future of social work dependent on continuous government funding.
I think that you will be able to find several responses to this question. One significant way that the creation of public work programs helped social work was that it helped to bring challenging predicaments into the open and public sphere. The notion of losing one's job and being unemployed was no longer seen as the source of private shame, but rather was appropriated by the government and seen as an issue where a public response could be warranted. This is fairly significant as much of social work seeks to overcome this particular barrier in terms of privacy and keeping truths concealed to a realm where there is a public reclaiming of the problem, empowering the individual.