It's not really a race to the bottom though. People will be payed what they're worth to a company. If that wage is not high enough, the person can go elsewhere or a competitor will come in a scoop those employees up and pay them better. People aren't enslaved to their job, they have the freedom to stand up and go somewhere else. It may not be easy, but that's what you have to do sometimes.
An employer has a vested interest in keeping his employees happy. If they're not happy they leave and the quality of service goes down. The quality of service goes down, so do profits. This mainly applies to small businesses (a majority of businesses by a significant margin). Not every business is trying to screw you.
Your point might stand when it comes to no skill jobs though. People need an incentive to get a skill and apply themselves. A $15 minimum wage won't do that. You're just going to have less people employed or people working less hours in those lesser/no skilled jobs.
I've been there making minimum wage. It fucking sucks. The work sucks and the pay sucks. My incentive was to get out and actually have nice things, and it turns out getting a girlfriend (and keeping her) requires money. Running a cash register 8 hours a day is something I don't even wish upon my enemies.
Research also shows that if you want to join the middle class in the USA, there are three things you need to do.
This is basically what I did. I don't have a college degree, but I just bought my first home. I'm making around $70k (for a small business) in a really nice area just north of Indianapolis. I'm now 29 and have a 2 year old and I actually love my job.
It takes dedication, and some luck. Opportunities are out there.