Basically the idea is the minimum wage reduces employment (http://modeledbehavior.com/2010/10/12/what-the-new-minimum-wage-research-says/). Another criticism is that the benefits don't necessarily go to someone who needs the help (for example it may go to teenagers getting their first job). I find it hard to understand having a good grasp of economics and having an intuition that minimum wage laws are good. Are price floors in general a good idea? At the very very least, they're a very kludgy and indirect method of helping poor people.
My understanding of the evidence is that minimum wage laws is that they are not especially harmful, but they're not especially helpful either (I have this link in my notes from when I did a bit of digging on this topic for the same reasons as you but from the opposite direction http://www.jstor.org/pss/1942818). Basically, minimum wages aren't a huge problem, but if your goal is tp help poor people, there are better ways to do it (for example the Earned Income Tax Credit seems to work pretty well).
Are price floors in general a good idea?
Imagine there are 100 manufacturers of cars, who discover that a byproduct (lets call it weberfoam) of the process of making cars is useful in making planes. Each manufacturer makes between 1 and 2 tonnes of weberfoam, of varying quality per month. 10 plane manufacturers want weberfoam, because it makes planes lighter. They're willing to pay quite a bit for it (up to $1,000 a tonne) but at that price will want only 1 tonne each. At $1 a tonne they'll buy 2 tonnes each.
What should the car manufacturers do? Would ...
The concept of minimum wage is one I'm rather attached to. I have dozens of arguments for why it helps people, improves the world, etc. etc. I suspect this view is shared by most of this community, although I haven't seen any discussion of it.
I don't have much understanding of the harms that minimum wages cause; and at what level of minimum wage those harms become relevant (ie. a minimum wage that would not be a living wage even working 24 hours a day is unlikely to have any of the same problems that a minimum wage sufficient to buy an aircraft carrier an hour would have)
So what are the harms that such laws cause?