I think you'll find those who support minimum wage laws also support the enforcement of those laws. I'm not sure if you're advocating for deportation, restrictions on immigrant labor, or something else entirely, but given that you're using terms like "illegal aliens," I'm going to assume you're not a fan.
We've already got mechanisms to crack down on black-market labor. Tax-dodging, not paying workers minimum wage, and subjecting employees to unsafe work conditions should be punished as such. In addition to punishing employers who exploit immigrant workers by paying them illegal wages, we should have much larger programs to allow unskilled undocumented immigrants to legally participate in the above-ground economy so the legal citizens aren't undercut. It's the same reason why we don't allow employers to pay teenagers below-minimum wage. Sure, teenagers don't have to support themselves (let alone a family) and might be happy to work for less, but if employers are allowed to pay kids five bucks an hour, they'd only hire kids and the minimum wage would completely fail.
In short - the only way to reduce black and gray markets is to punish the exploiters and decriminalize the exploited. Punish the pimps instead of the prostitutes, the drug-lords instead of the addicts, and the business-owners who skirt sensible labor regulations instead of the immigrant workers.
I'm not the parent, but as a second-generation immigrant, I broadly agree with all of this. I'm fine with whatever laws we all decide, and I think they should then be vigorously enforced.
Our current wink-wink scheme, supported by both major parties, is bad for a lot of people, and in many way most harmful to illegal aliens (who are abused and/or are virtually enslaved).
You think literally millions of people are sneaking across the southwest desert and risking having their children seized and put in cages[1] to be "abused and/or virtually enslaved"?. No, that's not how it works. Immigrant jobs tend to be poorly regulated (no taxes witheld, benefits delivered nor social services funded, etc...). But they're jobs and they pay what jobs pay, generally around minimum wage, because that's where the market chases.
I'll leave the both-sides-do-it argument. It's wrong, but there's no way to argue that on this site without getting myself banned.
[1] A policy very much not "supported by both major parties"...
I think that most illegal immigrants don't fully understand the downsides, yes. But whether or not that's true, I think it's unethical for us to tacitly encourage them to put themselves and their children in a position where they can be practically enslaved, or even raped or murdered, with little recourse to law.
> You think literally millions of people are sneaking across the southwest desert
No, even without the later qualifications; unique illegal border crossers aren't that high, and aren't all “sneaking across the southern desert”. It's not 2000 anymore.
We've already got mechanisms to crack down on black-market labor. Tax-dodging, not paying workers minimum wage, and subjecting employees to unsafe work conditions should be punished as such. In addition to punishing employers who exploit immigrant workers by paying them illegal wages, we should have much larger programs to allow unskilled undocumented immigrants to legally participate in the above-ground economy so the legal citizens aren't undercut. It's the same reason why we don't allow employers to pay teenagers below-minimum wage. Sure, teenagers don't have to support themselves (let alone a family) and might be happy to work for less, but if employers are allowed to pay kids five bucks an hour, they'd only hire kids and the minimum wage would completely fail.
In short - the only way to reduce black and gray markets is to punish the exploiters and decriminalize the exploited. Punish the pimps instead of the prostitutes, the drug-lords instead of the addicts, and the business-owners who skirt sensible labor regulations instead of the immigrant workers.