Most minimum wage workers are not poor by any standard. Stating this fact makes me a hateful s.o.b., I know, but even such clever rebuttals as that don't make mandated wage increases for the lowest paying jobs any kind of tool for fighting poverty.
If the minimum wage peddlers really wanted to help low-income workers, they would push for for expanding the state earned income tax credit. That way high school students wouldn't be the primary beneficiary of the poverty-busting crusade. The EITC can be targeted to families. Since there are so few of them who actually depend on the minimum wage, the public costs would be much smaller and the desired effect would be achieved.
It's a win-win unless your real goal is just politics.