Does minimum wage help the poor?

A minimum wage makes it illegal to employ workers whose marginal product is less than whatever the real value of the set nominal wage is. That creates a surplus of workers - unemployment.

From what I gather from economists, the deep solution to combating poverty and equality is a negative income tax (or variant of). Rather than transfer money from employers of minimum wage jobs to employees who have minimum wage jobs, a negative income tax transfers money from all tax payers to all people who are poor. This makes a negative income tax both fairer and less distortionary than a minimum wage.

Roughly half of economists agree (though it varies according to the survey) with the above story, and as such, are against a minimum wage. The other half believe the story, while correct, is incomplete, and complicating real world factors open up the possibility that a minimum wage could be somewhat effective in certain scenarios. Daniel Klein conducted a survey of economists' opinions on minimum wages here, where these were the most common reasons given in favor of a minimum wage:

A. Equalizing an imbalance in bargaining skills (i.e., bargaining experience, articulateness, confidence)
B. Inducing employers with monopsony power to increase employment by their firm.
C. Inducing a transfer from employers to (generally less well off) workers, albeit with possible small disemployment effects.
D. Coordinating the low-wage labor market by making it common knowledge that jobs pay at least $7.25.

Economists against a minimum wage contest each of these points, and furthermore, accuse their economic opponents of politically and ideologically motivated opinion forming. In some cases, the opponents do not even deny it, one describing a minimum wage as "A low cost demonstration of concern for low wage workers that causes little damage. Elicits a buy-in by low wage workers to the polity". Critical commentary here, here, here, and here.

As a non-economist, it seems plausible that a minimum wage might have a small positive benefit in certain scenarios. The evidence is however underwhelming (it echoes to me of climate skeptics who say CO2 won't actually cause warming due to a complicating effect like a negative feedback - when in doubt I put my money on the simpler explanation). The larger point really seems to be the effect of a minimum wage is at best quite small, which is part of the reason its effect has been difficult to empirically measure. Those who tout the minimum wage most strongly, which are the politicians and union leaders, are clearly after securing political points rather than a real solution to poverty and inequality.

A final note: before I researched this I assumed there was an economic consensus on this issue, and in fact, according to Harvard's Greg Mankiw here, 79% of economists agree that "A minimum wage increases unemployment among young and unskilled workers", which is why Greg includes it in his list of top 14 public issues that economists agree on. However, he says here that "One issue that fails to generate consensus is the minimum wage: 37.7 percent want it increased, while 46.8 percent want it eliminated.". While these statements might be logically compatible, they leave non-economists like myself a bit baffled. No wonder the public end up believing whatever the influencers want to tell them.

Like all price floors it creates a surplus; it necessarily creates unemployment. I don't know how much, but it can't be helping.