By this logic, Republicans can never win an election. The Democrats need only offer $20 an hour minimum wage, or $50, or whatever it takes, to where the Republicans can't stomach the notion of economic suicide for the country.
People earning the minimum wage may want it to go up, but they may also want to keep their job. At some point, those two wants cannot both be met. Republicans should focus on alternatives such as more opportunities for low-wage workers to advance in the ranks, to learn new skills so as to land better jobs, and expanding the earned-income tax credit so that those who work, even if for little, get a boost to where they're better off than if they just gave up and tried to get themselves classified disabled or something.