I know it’s terribly old-fashioned to ask whether any power the federal government wields or proposes to wield is authorized under the Constitution, but let’s ask about the constitutionality of minimum-wage laws anyway.
Article I enumerates the legislative powers Congress is to have. It says nothing about setting minimum-wage rates, nor, generally, that Congress is allowed to dictate the terms of contracts between individuals. If someone had asked one of the Founders after the Constitution was written, “Does this new plan of government allow federal politicians to tell us how much we may charge for our goods or our labor?” the answer any of them would have given would have been, “Of course not!”
In this piece, Mike Maharrey of the Tenth Amendment Center addresses the claim that minimum-wage laws were contemplated under the power to “regulate commerce among the states.” He cites a pretty good authority, James Madison, that the Commerce Clause was not meant to confer plenary economic power.
Initially, the Supreme Court ruled against the federal minimum-wage law, but after the 1937 switch by Chief Justice Hughes to the “liberal” side (that is, statist and illiberal), the Court upheld it, along with everything else FDR dreamed up to subject the country to control by Washington, D.C. Alas, that is a legal blunder that will never be undone.