One can support minimum wages and still understand that they raise unemployment for those subject to them. It’s far from settled whether there is a net benefit or not, but it’s pretty much universally accepted that they do raise unemployment.
The easy proof is to consider a minimum wage of $50/hour. One consequence of this would be the near complete elimination of the entire retail employment sector. Automatic electronic clerks would be installed at every store to take money. The budget for technological improvements to self-service retail technology would be huge.
Obviously, employment is decreased as millions of positions are eliminated. Funding into technology to replace those positions, however, is massively increased. Engineers enjoy full employment, as do technicians to service these machines. These increases in high tech employment would not come close to offsetting the massive lower end job losses though. There would also be massive frictional losses as companies struggle to retool quickly. In the long run, there may even be a net gain to society, as new technologies to replace human workers are developed.
But one incontrovertible fact remains, the higher minimum wage caused higher unemployment. Not everyone can be an engineer or a highly skilled worker. Minimum wages have the strong potential to hurt the very people they are designed to help, eliminating their jobs, replacing them with machines. Something to consider the next time you pump your own gas (and pay yourself at the pump), use an ATM, or a self-service checkout at the grocery store.