The issue is not about employees not getting their earned wages from their employer. If that ever happened, the employer would be subject to a greater liability payment than what was originally due and is a fairly straight-forward labor case.
The whole meme and joke around the $15/hr proposed minimum wage increase is that the current Federal minimum wage is $7.25, so increasing to more than double to $15/hr is an unsustainable increase rather than the usual gradual increase, like first increase to $8-9/hr, then after a few years increase to $10-12/hr and so on. A company will always pay the employee the least they can which is directly proportional to the employee's role within the company. If you work an entry-level part-time fast food worker job, which requires no experience, no training and no quota, you would of course get paid the minimum wage. But if you work as the assistant manager or manager role, which does require some experience, some training and some sales quota, you will get paid much more.