I think it's hard to view in terms of timelessly constitutional or not; the Court's favored interpretations of bits of the Constitution have shifted back and forth widely since the Supreme Court was established.

So I suspect the strongest possible statement is that this is a program most people accepted as constitutional without much controversy at the time of the New Deal, but since then the Court's interpretation (and the political climate in the country, of which the Court is for better or worse a reflection) has shifted.