Employers have a responsibility to pay their workers a living wage. Responsibility is the defining characteristic of conservatism.
A living wage will get hundreds of workers off welfare. Reducing welfare is a goal of conservatives.
A living wage enables workers to be independent of government aid. Rugged individualism is the ideal of conservatives.
Raising the minimum wage to a living wage will give workers more money with which to purchase goods and services. This is good for business, a cause advocated by conservatives.
I realize that there are small businesses that cannot afford to pay their workers a living wage.