You're right. From my understanding of junior macroeconomics, one problem is that even if you do increase the minimum wage, you're probably subjecting your society to two mechanisms: long term inflation (positive/negative depends on society) and structural unemployment. As businesses seek cheaper labour, this would result in greater law evasion by increased migrant workers to meet labour demand or outsourcing work, which result in greater loss of domestic jobs.
Of course, I haven't taken senior economics, which probably places greater emphasis on mitigating negative externalities through government intervention, so I'm incorrectly oversimplifying the issue.
This reminds me that high schools should incorporate basic micro and macroeconomics into the curriculum.