Minimum wage was meant to be a living wage. Here's FDR's inaugural speech about minimum wage (bolding added by me):

"In my Inaugural, I laid down the simple proposition that nobody is going to starve in this country. It seems to me to be equally plain that no business which depends for existence on paying less than living wages to its workers has any right to continue in this country. By "business" I mean the whole of commerce as well as the whole of industry; by workers I mean all workers, the white collar class as well as the men in overalls; and by living wages, I mean more than a bare subsistence level-I mean the wages of decent living. Throughout industry, the change from starvation wages and starvation employment to living wages and sustained employment can, in large part, be made by an industrial covenant to which all employers shall subscribe."