A straight across minimum wage of $11 an hour would likely do more harm than good as many small businesses would choose to cut jobs rather than pay the higher wage.
However, I think the minimum wage should be tiered to reflect the differences between businesses and their available resources. The small business with 50 employees or less should be able to remain at the current levels, as long as that business owner's compensation does not exceed between 100 or 150 times that of the minimum paid worker.
But the large businesses should be required to pay $11 an hour or even $15 an hour as a minimum depending on their executive to minimum compensation ratio. If their CEO is getting paid over 150 times or more than what the minimum wage worker is currently making, they have to boost the minimum wage at that company to $11 an hour, and also potentially face higher corporate taxes -or- reduce the executive pay to a level below the 150 times minimum compensation mark to avoid this.
If the CEO is making over 250 times what the minimum wage worker is currently making, they have to pay $15 an hour, and potentially face higher corporate taxes -or- reduce the executive pay to a level below the 150 times minimum compensation mark to avoid this. This would entice boards, shareholders, and executives to ensure that ratio doesn't go higher than the 150 times level, and protects small businesses from having to do something they don't have the resources to do.
Corporations today have more cash than they know what to do with. But unfortunately the way the tax code and our accounting systems are currently set up, paying workers more is not seen as a good use of money. But burning the cash by spending it on stock buybacks is.
Thomas Piketty has pointed out some intriguing data on this problem, lets use logic to solve it. In the scenario above, I believe, everyone wins.