AFAIK most jobs that would pay under what's enough to live on are entry level jobs usually carried out part-time by students. Denying them such job is removing them that source of income - and in case of jobs that require experience denying them a chance to gain that experience. This means you are not reducing any social externality. You are just shifting those externalities to a different set of people.
I'm also wary of "daling with" this kind of social externalities even if it was possible. If we apply your argument to drugs than the conclusions would be that alcohol and any other drug should be made illegal because it has a social cost. But I believe people still have the right to do whatever they want with their body and mind as long as there is no direct externality, and I'd rather live with any indirect externality that is the result of that.
This is a rosy view of things, but I don't believe its true. Walk into plenty of markets in poorer or even middle class areas and you will find older people doing these jobs out of necessity. I've seen 'entry level job' used very often as a way to exploit people with not much choice.
The situation is diametrically opposite - minimum wage forces businesses to engage in welfare. Those who have enough margins to engage in welfare might do it; the others may disappear with the jobs, or cut the jobs by reducing service/automating/outsourcing (again leading to the same "social burden" existing).
That's the negative externality.