A higher minimum wage, with its possible disemployment effects, is often counterposed with wage subsidy schemes like the Earned Income Tax Credit as an alternative for helping low-wage workers. But as Mike Konczal and Arindrajit Dube explain, it's better to think of them as complementary policies. Wage subsidies help low-wage workers, but also serve in part as a subsidy to low-productivity employers, and they have the benefit of inducing more people to join the workforce. Minimum wages (within reason) also help low-wage workers, but serve in part as a tax on employers, and have possible downsides in terms of discouraging employment. So you want to try to do some of both.
But I still think minimum wage regulations are far from optimal. The real policy mix you're looking for is a blend of wage subsidies (to encourage work) and something like a Guaranteed Basic Income program that just hands out cash to people regardless of what they do.
A GBI helps people by giving them money, obviously. It also serves as a kind of de facto minimum wage, since if people can earn money doing nothing, in practice you're going to need to offer them higher pay to get them to work. But it's much more flexible than a minimum wage. In a GBI world, an employer has to make work somehow appealing enough to get employees even though everyone's guaranteed a basic minimum whether they work or not. But that "appealing" factor could be high wages, could be valuable skills and training, could just be a pleasant work atmosphere, or could be some combination of the three. Current minimum wage policies sort of try to achieve these goals by having exemptions for educationally rewarding internships or vocational programs. But these exemptions manage to be simultaneously too prone to abuse and too inflexible to capture the full range of possible scenarios that arise in human life.
Obviously, real-world politicans and activists operate within constraints, but when I'm dictator that's what we're going to do. No minimum wage rule, no sloppy enforcement, and no weird loopholes. But everyone gets a check every month, and employers need to offer a proposition that's more attractive than leisure even though pure leisure won't necessarily lead to immiseration. I'd also add in a progressive payroll tax that's somewhat negative at low levels to prevent the disemployment effects from getting out of hand.