Setting a minimum wage is not a legitimate government function. Government can no more determine appropriate wages than appropriate selling prices. Only an employer and employee can legitimately determine appropriate compensation for services rendered and only a buyer and seller can determine appropriate prices.
What does a government enforced minimum wage do? 1) It decreases the number of available jobs. 2) It places an artificial value on labor which should instead be determined by supply and demand. 3) It increases the cost of doing business and increases prices.
If you are legitimately concerned about employees earning to little I have a suggestion. When you buy groceries offer to pay 25% more for the groceries than they sell for. Tell the store to distribute the surplus among the lowest paid employees.
The Great Weight of History is against you. It all started in Babylonia. Gov't has always had to regulate business. In "Libertarian World", gov'ts aren't supposed to do this, but that is "theory". The real world works differently. Put down Ayn Rand for a while dude, and read something like "Common Things", or "Nickle and Dimed in America", or "Chasing the Red, White, and Blue." parsy.