There were two main sources of criticism for the New Deal. There was criticism from the right and criticism from the left.
The New Deal tremendously expanded the scope of government intervention in the US economy. It made the government more involved than it ever had been before. Not surprisingly, this caused a great deal of anger from people who were staunchly conservative. To them, the New Deal was essentially socialism being imposed on the United States. They were very upset at what they saw as the introduction of socialism.
However, the New Deal also came under attack from the left. The New Deal, people on the left felt, did not go nearly far enough. There were many people who advocated for more radical changes to be made. There was, for example, Dr. Francis Townsend. He wanted the government to give really large pensions to all elderly Americans so as to stimulate demand. Huey Long in Louisiana had a “Share Our Wealth” proposal that would have redistributed income on a massive scale. These were much more radical proposals than President Roosevelt was willing to support.
Thus, the New Deal was attacked from both sides of the political spectrum.