I believe you have to look at the minimum wage question like this.
If minimum wage workers are paid more then the work they do has to be passed on to what they produce weather it be grocery produce, clothes, cleaning or factory produce.
Imagine if higher paid workers have to pay a bit more for the above items or services wages are only part of the cost of them.
Countries other than the US do this and are very successful in providing a working wage for all. Just like communism.
There is a problem with this approach in that it will make a country less competitive and some may loose their jobs from lack of exports and an increase of imports. Some economic sectors will not be effected such as the tourist sector because their jobs can't be exported .
There is no point in competing with say China on wages each country should compete on their natural advantage.
I'm not an economist just my thoughts.