Until the Great Depression, minimum wage laws were consistently struck down by the Supreme Court. Using the 14th amendment as their basis, the supreme court's position was that the government did not have the power to regulate contracts between employer and employee.
The Great Depression caused them to take a fresh look at this and the result were state minimum wage laws.
Over time, the federal government itself got into the act and now we have national minimum wage laws and state ones as well.
The constutionality of these is murky. I personally don't agree with minimum wage laws. I don't think the government has any business telling me how much I should be paid or how much I should pay someone else. But I do think that it is within the power of the states to pass such laws by their duly elected representatives.
On the other hand, I don't see how the federal government has the authority to establish national minimum wage laws. That seems to me to be blatantly unconstitutional. The Supreme Court "found" this sweeping power in the Constiution in the clause on the federal government having the power to regulate interstate commerce. But that's really a cop-op in my opinion.
The problem with minimum wage laws is that they're arbitrary and pointless feel-good measures. In today's global economy, minimum wage laws might as well be called outsourcing laws or automation laws. If you're making minimum wage, that means that the value of your talent and skills is so low that you need the government to step in and artifically inflate it.
I remember some years ago having a debate on Usenet about minimum wage laws around the time they had last raised it and I said "I predict that in 10 years, you'll see a lot of these jobs being automated or sent overseas. So if you find yourself checking yourself out at the grocery store in a self service line, remember this." Of course, back in the early 90s, I got flamed and patronized on the reasons why grocery store check out lines couldn't be automated (impossible to stop shop lifting was the basic argument).
But sure enough, we have seem automation and outsourcing take their toll and the very same people who argue for minimum wage laws are the ones who complain about the results -- as if there is no connection between the two.
That said, I do believe states have the right to enact minimum wage laws. I don't see anything in the constitution that says (or implies) that the government can't do it. I just think it's a bad idea.