I originally wrote this post based on the view that that was the position you were putting forward; having reread your post I'm not sure it is, but here's an attack on that philosophy anyhow:
I think that it's based on the fallacy that offering people a job under poor working conditions, *which they are free to take or leave*, is harming them. It's not, it's actively helping them, more than not offering them a job is, it's just not helping them very much.
If someone offers me a job screwing lids onto toothpaste tubes for 20 hours a day for thruppence hapenny, they haven't hurt me in the slightest, because I can just turn them down. A job on less than a "living wage" is still often better than no job at all.
The actual reason a minimum wage is a good idea is not nearly so simplistic, and doesn't make a good slogan.
An economic transaction occurs when, and only when, I have something (in this case labour, but it could be anything) which is worth more to you than to me. You give me an amount of money somewhere between it's value to me, and its value to you, and both of us gain. If you offer less than it's worth to me, I just turn your offer down, and obviously you won't offer more than it's worth to me.
That region between my goods' value to me, and its value to you, is called the "core" of the transaction, and how it gets split is hard for economists to model. The splitting of it determines how much of the benefit of the transaction comes to me, and how much goes to you.
The benefit of a minimum wage is that it ensures that more of the core of an employer-employee transaction goes to the employee, and less to the employer. The potential disadvantage is that if you set it above the value of labour to the employer - that is, outside the core - then they'll stop employing, and both they and their would-be employee suffer from losing out on a potentially mutually beneficial transaction. But while it's inside the core, it's doing good, and empirical evidence suggests that the impact on employment figures of minimum wages is not large.
There's absolutely nothing immoral about offering to employ people in appalling conditions, *provided* they can just turn you down. The argument for a minimum wage is *not* that it's punishing wicked exploitative employers, it's that coercing employers who are already helping the poor a little by offering them jobs on poor terms to help them still more by offering them jobs on less poor terms is a good thing.