This is in fact their purpose. Minimum wage laws destroy low-paying jobs by making them illegal, preventing the least competitive members of our society from having the hours and days of their lives "mined" by an employer for negligible compensation.
That having these laws "reduces welfare" is, I believe, a conclusion not supported by fact.
A living example can be found in urban Brazil. By failing to outlaw and enforce certain minima (building codes, wages), large Brazilian cities have created vast marginal neighborhoods that no one wants to live in.
Based just on the example of favelas alone, I would argue that having laws to guarantee minimum wages is one thing a government can do immediately to protect the weaker members of a society.
I think you have not addressed another important duty of governments: to provide a reasonably rigorous educational launchpad so that the less fortunate need not always remain so.