The run away inflation scenario you describe would only happen if the money supply grew as exponentially, which no competent central bank will allow to happen.
To be sure, minimum wage will contribute to inflation, but not in any catastrophic sort of way. Earning minimum wage does not mean a person will suddenly be able to afford to pay double for basics such as food, healthcare, transport etc. Earning at that level pretty much just means a person makes just enough to scrap by and not starve or die in the cold.
The goal of minimum wage is not to make anyone rich, rather it is to ensure a minimum standard of living among those who work at the lowest rung and the money for this increased payroll to come from corporate profits.