The minimum wage hasn't been raised in several years and isn't enough to support more than one person above the poverty level. As more than one quarter of those paid the minimum wage have children, Economic Policy Institute -Raising the Federal Minimum Wage to 10.10, that is a significant problem. Fortunately, proposals to raise the minimum wage are gaining traction.
Leaving aside those who have philosophical objections to government, the argument against raising the minimum wage is that (i) raising the minimum wage harms poor people by destroying their jobs ("Raising The Minimum Wage Is A Job-Killer) and (ii) most of those who earn minimum wage are not in fact poor so raising the minimum wage doesn't help the poor ("People Who Make Minimum Wage Aren't Poor").
These arguments are somewhat contradictory; it is possible, however, for both arguments to be true so both arguments should be addressed, even if those who make these arguments are insincere.
People Who Make Minimum Wage Aren't Poor, So Raising The Minimum Wage Doesn't Help The Poor
In evaluating this argument, the first question we should address is the truth of the statement that most of those who earn the minimum wage aren't poor.
It is apparently the case, as conservative economists have noted, that only 11.3% of those making minimum wage live in households officially defined as poor. See Most of the Benefits of a Minimum Wage Increase Would Not Go to Poor Households. On the other hand, approximately 23.1% are in households earning less than $20,000 per year, just barely above the poverty line. Economic Policy Institute - Raising the Federal Minimum Wage to $10.10 - Figure H. The same (admittedly liberal) source notes that another 28.8% of workers earning minimum wage are in households earning between $20,000 and $40,000.
To be fair to the conservatives, they appear to be technically correct in the first part of their argument. It should also be clear, however, that a much larger number of those affected are very close to the poverty line even if not technically poor.
More importantly, the conservative argument is fundamentally flawed: the relevant question is not what percentage of the benefit of a minimum wage increase goes to the poor, but what the effect of such an increase is on the poor and the poverty rate. Fortunately, there is data here too. A study of the effects of changing the minimum wage on poverty (based on the effects of prior changes) suggests that raising the minimum wage to $10.10 will cut the poverty rate among non-elderly American from 17.5% to 15%. See Minimum Wages and the Distributions of Family Incomes, p.33.
This study acknowledges that there are more efficient ways to reduce poverty - such as expanding the Earned Income Tax Credit to those without children. If conservatives want to put that on the table as an alternative to raising the minimum wage, I'd be willing to consider it. I'm not holding my breath though.
Raising The Minimum Wage Is A Job Killer
The other argument against raising the minimum wage is that it destroys jobs. Here again, the conservatives are right. A 10% hike in minimum wages is estimated to cost between 1% and 3% of minimum wage jobs. See Most of the Benefits of a Minimum Wage Increase Would Not Go to Poor Households. Raising the minimum wage to $10.10 therefore would cost between 4% and 12% of minimum wage jobs.
Of course as noted above, even after accounting for the job losses imposed by minimum wage increases, the increases reduce poverty significantly. There is also evidence that raising the minimum wage creates jobs by increasing GDP. See Economic Policy institute - Raising the Federal Minimum Wage to $10.10 - Appendix 2. The job creating effect would be enough to cancel out the job destroying effect, if the lowest estimates of the job killing effect are correct. Of course, even in that situation, those who lose jobs will not always be the same as those who gain them.
At the end, therefore, what we know is that raising the minimum wage will help poor people over all, but badly hurt a small fraction of poor people.
Conclusions
1. We should raise the minimum wage to $10.10, which will help to significantly reduce poverty.
2. Minimum wage increases should be phased in over time, like the current proposal to raise the minimum wage to $10.10. This gives a better chance for those who lose their jobs due to the minimum wage increase to find new ones quickly.
3. Most historical minimum wage increases were on the order of 10% or so per year, like the proposal to increase to $10.10. See Raising the Federal Minimum Wage to $10.10 - Appendix 3 Raising the minimum wage more quickly might exacerbate the negative effects.
4. Raising the minimum wage to $10.10, would restore the minimum wage to its highest real level and enable a minimum wage earner working full time to support a family of three at a bit over the poverty level. See Raising the Federal Minimum Wage to $10.10 - Appendix 3
5. Once the minimum wage is high enough that working a minimum wage job is enough to lift one above the poverty level, further increases will have a smaller effect on poverty. Thus proposals to raise the federal minimum wage beyond $10.10 will likely have less of a positive effect on poverty.