The minimum wage is a cynical wedge issue. If political "leaders" who support raising it really cared for the poor, they would instead do something about monetary inflation. If we got rid of the Federal Reserve's monopoly and allowed free competition in currency, then consumer prices will be subject only to the laws of supply and demand and not to the shrinking value of the dollar.
Meaning, if the monetary unit's value remained stable year after year, the cost of living won't rise year after year and there'd be no reason to raise the minimum wage.
But that's not the main reason minimum wage laws are wrong, which is, they're immoral. They impose a coercive, arbitrary, one-size-fits all standard on employee compensation, when employer and employee might prefer alternatives. Such alternatives could be free meals, more flexible schedules, more comprehensive health coverage, or other, more informal arrangements.
It's a question of two consenting individuals agreeing to the terms of labor and the terms of compensation. Nothing in the Constitution gives Congress authority to interfere in that relationship, and no state or local legislature has any moral authority to interfere either. It's literally none of their business.
This should be understood implicitly by everyone. It's been accepted by most people for a few decades at least that the State shouldn't interfere with sex between consenting adults. I agree. But we can also acknowledge that frequent outcomes of this include unwanted pregnancy, divorce, jealousy-induced violent crime, and transmitted disease.
If sex isn't The State's business, then how do other consensual behaviors, that don't create such problems, suddenly become the State's business? Just as The State shouldn't interfere with your relationship with your loved one, it also shouldn't interfere in the relationship between the hirer and the hired.