The other thing is that 'we live in a globalised world' - of course fast food and hotel rooms need local staff, but pretty much everything else (manufacturing, call centres, design) can be outsourced overseas (doesn't matter which seas here - it applies as much to the UK as the US).
Min. wage increases drive that, in the age of multinationals, and even small businesses can easily do it.
Over time, in theory, the world will run out of places to get cheap labour, and then who knows.
I'm not against min. wage, but it isn't a straight up solution. Should hard working stupid people get ~$20 an hour (or whatever), vs the CEO getting, I don't know, $1k? Of course it's fairer if the CEO only gets a certain multiple of the lowest paid. But life isn't fair.
IMHO getting shot of (in the US) the craziness where 'part time' employees miss out on a whole bunch of stuff (healthcare) is the first, and more important, step. Big companies will always work to minimise costs. Stupid laws (or clever managers?) mean loopholes.
People shouldn't slip through cracks, and be taken advantage of because they *need* a given job. (But then, I believe in universal healthcare, and not forcing people to wreck their backs because they are treated like robots at Amazon facilities...)