Yes and no.
I do not support raising minimum wage to $15/hr. Labor is a commodity/product and you cannot artificially set the price of a product, the market determines the cost.
However, I do support raising the minimum wage to 11.50. People working minimum wage are below the poverty line and because of that are eligible for government assistance. Government assistance for people that actually work costs the American taxpayer approximately 7 billion dollars a year. To get those people above the poverty line and off gov't assistance the minimum wage should be raised to $11.50 an hour.
Companies that can not pay their employees 11.50 an hour will just get eaten up by larger companies that can. Thats how business works.
If you do not support raising the minimum wage to 11.50 an hour, what you are effectively saying is that you support supplementing the profit margins of multi-billion dollar multi-national companies with your tax money, These companies know the effect of their wages on the economy and make a point to educate their employees on how to get on gov't assistance. It's another form of corporate welfare.
You are ok with small businesses getting eaten up by larger companies because thats how business works, but you won't stand for multi-billion dollar companies doing the same thing only on a larger scale?