In a functioning state with basic welfare, the minimum wage is important. It prevents employers from getting significant work from employees while also having the employees be making low enough income to qualify for welfare. When this happens it means that tax payers are basically paying for the work of the employee for that company instead of the company paying for it.
In theory, the minimum wage should prevent this. In practice, it still happens quite a bit in the US. This is a reason the minimum wage in the US is currently broken, and at least in some areas where the cost of living is high, the wage needs to be raised. As you have people working 35-40 hours a week or more. Almost, at, or beyond full time job levels, and still making low enough to qualify for food assistance.
This isn't limited to fast food or retail jobs either. Even as highly thought of occupations as airline pilots fall into this category. One common situation among newly graduated airline pilots is being hundreds of thousands of dollars in debt because of their training, and making wages low enough to qualify for food stamps in their first years flying as copilot.
Some occupations don't even have the minimum wage applied consistently and have to fight for it to be done if they end up paid less than it. Company truck drivers are notorious about this. Often paid by distance traveled, poor luck, bad conditions, or bad dispatching by the company can leave a truck driver in the best case, with a meager stipend for 24-48 hours of sitting, and in the worst case, completely unpaid for a bad weather event or traffic blockage.
The Walmart class action lawsuit accused the company of not paying its truck drivers for certain activities including inspections, waiting at weigh sales, rest breaks, washing trucks and fueling. Judge Illston’s ruling was that truck drivers should be paid at least minimum wage for “all the time they work.” The Walmart class action lawsuit could result in truck drivers receiving more than $100 million in back pay.
The point is that while currently broken in many areas, a proper minimum wage setup should be enforced, because if you're not working for the minimum amount needed to survive then you might as well not be working at all or giving your work away on a non-paid volunteer basis. If we expect to live in a society where we want to guarantee people even the most basic welfare as nutrition assistance, just keeping people from completely starving (often not even keeping them from being hungry, just not dying from it) then we need a minimum wage in order to keep the welfare from going to the companies instead of the people. Because that's exactly what happens when people are being paid below a proper minimum wage.
There are other reasons too, of course, but this is the one I've always found most compelling and easiest to logic out to people who were against it. The fact that no or too low of a minimum wage costs tax dollars tends to get to people. And while some people are against welfare as a whole, many people are less against the idea of helping people to not starve at a minimum.