Don Boudreaux asks a good question:
Why do so many people who weigh in on this issue make such fantastically illogical "arguments" in favor of raising the minimum wage?
Here are my theories:
1. Tribalism. A minimum wage is a totemic issue, that separates "us" and "them". And when your side's under attack, you rally to defend your totem.
2. The cluebat's weak. The job losses caused by a minimum wage aren't large enough to show up in the macroeconomic numbers that get all the newspaper headlines. This is because the minimum wage is low enough not to have a big impact on wages, and the price-elasticity of demand for labour is low. Sure, if you look carefully there is evidence of cuts in jobs and hours (pdf), as theory predicts. But it's easy to avoid this evidence, especially if the confirmatory bias means you don't look in the first place. And it's easy to find people who have kept their jobs and hours, and so benefited from the minimum wage.
3. Exploitation. What angers some leftists about low pay is not poverty, but the fact that some people are profiting from it. So they support a minimum wage as a way of curbing the power of the rich, not as a way of enriching the poor. They think extreme exploitation is such an evil that unemployment is better.
4. Reaction to market imperfections. The labour market often just doesn't function like a basic textbook, well-clearing market. This encourages the inference that intervention in the market might be beneficial.
5. A transitional demand. When I was a Trotskyite in the early 80s, I supported a minimum wage precisely because it was "impractical." We did so to show that capitalism* could not deliver living wages for all.
I'm still not sure if this argument is wholly wrong.
6. The counterfactual. The minimum wage (along with tax credits) is pretty much the only game in town for reducing poverty.
But there is an alternative - a citizen's basic income. Intelligent supporters of a minimum wage acknowledge that a basic income is a better way of helping the poor. But the problem is, the alternative to a minimum wage right now is not a basic income, but nothing. And some think a minimum wage is better than nothing.
It's here, I think, that many libertarians are missing a big trick.If they were to call for simple, relatively unintrusive redistribution - say a flat tax plus a basic income - they could show that they were genuine about wanting to reduce poverty, whilst highlighting the irrationality of statist intervention.
As it is, rightist libertarians' silence (or hostility) to redistribution allows tribal defenders of the minimum wage to claim the moral high ground of being on the side of the poor.
* Vulgar libertarians please note: by capitalism, I mean private, unequal and hierarchical ownership of capital, not a free market economy.