How much does the minimum wage help the poor?

Not very much (JStor via MarginalRevolution):

About 35 percent of the total increase in after-tax benefits goes to families with income less than two times the poverty threshold, a common definition of the working poor or near-poor; nearly 13 percent goes to families principally supported by low-wage workers defined as earning wages at or below 117 percent…of the new 1996 minimum wage; and only about 14 percent goes to families with children on welfare.

Unlike most public income support programs, increased earnings from the minimum wage are taxable. Over 25 percent of the increased earnings are collected back as income and payroll taxes…Even after taxes, 27.6 percent of increased earnings go to families in the top 40 percent of the income distribution.