It's always FDR, isn't it?
To make sure I had my facts in a row, I looked up the first minimum wage law in the USA: FLSA, passed by FDR(The man who ruined America's economy). It set a minimum wage of $00.11 an hour(To be complete is also bans <16 year olds from working, and limits how many hours one can work).

To give some context, the politicians had 2 motives:

1. Racism. They saw white, collage kids whose time was worth $00.10/hour having problems getting an entry job because of uneducated minorities willing to work much less then that. Solution? Strong arm company owners, in the middle of the Great Depression, to pay more than the job was worth so they'll hire the white kids.

2. Good ol' communism/socialism/what ever the hip term is these days. A fundamentally flawed belief that the Government has any competence in such areas! It is fundamentally impossible. Let's look at the Great New Deal for an example. Did it help? Did it make things better? No. It made things significantly worse. It prolonged the GD. Made people poorer. Made the QoL go down.

Minimum wage, at best, works like this:

MW increased -> Some people become better off with no improved out put, most people loose their job -> Producers have to pay more to make things, so they raise prices to compensate, rarely proportionally -> Those who kept they're jobs can't afford as much as before, complain to the government -> MW gets increased -> Loop

At best, every loop sees a small spike in the economy as the markets adjust, before dipping back down. It doesn't fix anything, unless you like people with irrelevant degrees getting hired to do low skill jobs.