Suppose I run a billion dollar company producing some commodity that everyone needs, and where most of my employees make minimum wage. If you raise the minimum wage by 10%, I raise the prices by that same 10%. Then our income goes up by 10% across the board, meaning that we pay our employees the extra 10%, but so do our profits. That means that my million dollar salary increased by $100,000. The situation of my employees hasn't changed a bit. But I can buy another Porsche.
Keep doing this - you force me to pay my employees more, so I raise prices, so the value of the dollar decreases. Over and over again. Only as the value of the dollar decreases, I get more and more and more dollars, while the regular people's savings becomes worth less and less.
Forget the minimum wage. Raise taxes on the wealthy. Make it pointless to hoard more money. Suppose my company has five million dollars to spend on salaries, and I get a million and the other two hundred minimum wage employees split the other four million. If my company makes an extra million, I give myself a half million raise, and give each employee an extra $2,500. That ain't nothing, but if I had to pay a 90% tax on my income over the first million (or first half million, even), there'd be no point in me keeping that money, and I could just give all that excess to my minimum wage staff, and help them actually get ahead. Or lower the prices on my product. And then there'd be some real public benefit. Instead of just giving the rich guy at the top an extra zero at the end of his bank account.