When minimum wage pays less than what a person can get on public assistance, one thing must change.
Aside from that, let's look at what jobs actually pay minimum wage:
Fast food. Janitorial. Parking lot attendants. Hospital housekeeping. Some security guards. Dishwashers. Physical manual low skill labor.
I'm not demeaning these jobs at all. The point is that they are entry level jobs which are mostly for people to acquire necessary skills to move up into a company. A proving ground to become management. As a young man, I hired entry-level employees as a manager of a national pizza restaurant chain. People either proved themselves and moved up, or they left. Nobody stayed at minimum wage.
Minimum wage is not meant to be a living wage, and I believe that if someone is working full time, they should be paid more than minimum wage. But this is not to say they should raise minimum wage.
Let's look at another apect: You say people should be paid more than what minimum wage pays today. Tell me what you think minimum wage should be? What's "fair"? And who decides what's "fair"
I believe the people can vote with their feet. If they're in a job and they're not happy with the wage, and they can't seem to get a raise, it's time for them to find a job that pays better, and move there. I worked for a company for 7 years, and after two years of voluntary wage freeze, they offered a 2% raise. I thought I was worth more, so I went looking, and found a place where I got a 35% increase, not to mention better working conditions and hours. I found it by God's grace, and I'm so thankful every day. But the way to move out of minimum wage jobs is to get a skill that business needs, and price yourself in the market.
Another by-product of increasing minimum wage is that the company should, to be fair, raise everyone else's wages that moved up in the organization. If I started at $8 and got a raise through hard work to $10, and then the government came along and said everyone should make $10, I'd expect my employer to give me $12. Can they honestly afford it, though? The best way to make more is to acquire skills and talents.