That is the standard conservative line. Every time Democrats suggest raising the minimum wage conservatives say it will increase the cost of labor and therefore drive down employment. I did a quick chart of the increases in the minimum wage since 1980 and compared the unemployment rates in the quarter before the rate was raised and the two quarters after. Here’s the chart:
The minimum wage has been increased nine times since 1980. The results show that the unemployment rate went up five times, went down once, and stayed the same three times. That would indicate that there is no iron rule that raising the minimum wage absolutely creates higher unemployment. (There is no doubt that the numbers do indicate a tendency to increase unemployment, but it is not a forgone fact as conservatives suggest.)
The most interesting numbers are GDP, which is a measure of economic activity. GDP actually increased six of the nine times the wage was raised. This would indicate that raising the minimum wage increases economic activity.
There are obviously many other things going on, but the bottom line is that raising the minimum wage does not automatically impact the employment rate.