No, not hardly. Also, some folks are not worth paying the existing minimum wage. Lack of skills and good work habits mean they cost more than they are worth. Makes it hard to learn skills and just plain learn how to work.
Minimum wage laws screw those folks as well as the skill-less who might be go-getters like many high school & college students. Toss in seasonal jobs and those same folks are priced out of the market by gov't. Then, toss in illegal alien labor competition and it gets even worse for young workers.
I'd say the illegal aspect is probably more corrosive to lower end wages for menial work than minimum wage. As MillCreek pointed out, it's not the biggest point of contention. I would concur with his point that disability is the new "welfare". Only drastically moreso. I know of more than a few people who gamed the system to be declared disabled. They did have health issues, but arguably could have found employment.
It's more profitable these days if you're making minimum wage range to go after benefits than employment. It's also useful in gaming unemployment numbers, and avoiding increases in welfare statistics. Social Security is the third rail of politics. Touch it, and die, unless you're padding the benefits. SSDI stats are grim, only 6% returned to work and 3.6% exited the program due to medical improvement. 1 in 14 workers is now on SSDI. Plus, you can get medical. Minimum wage rarely gets medical, and even more rarely can afford it.
The following is a pretty good presentation by Pennsylvania's Secretary of Public Welfare.
http://www.aei.org/files/2012/07/11/-alexander-presentation_10063532278.pdfSlide 8 is very very telling.
Slide 15 should be scary.
Basically, the number of private employees is essentially flat lined. Government employees and benefit recipients have increased. This is not good. You have less and less workers for every government worker or benefit recipient. Even with productivity gains, you want a wide ratio. We don't have it. Even ignoring libertarian ideology, you want a wide ratio of private workers to government workers or benefit recipients.