Both sides are guilty of sloganeering rather than careful analysis on this issue. Just to be clear, I favor raising the minimum wage to what it was, in real terms, in the '60's and '70's. It worked well then and would do so again. I also favor indexing the minimum wage to inflation, like everything else is these days.
First to the pro side. We hear that "you can't live on minimum wage". This is hogwash. I live perfectly fine on just over $1,000 a month, and this is less than the take-home pay for someone making minimum wage. (Don't forget the Earned Income Tax Credit kicks in.)
We also hear "You can't support a family on minimum wage". Well, duh! It used to be that people didn't get married and start a family until they were financially stable. We have completely lost the sense of personal responsibility and personal character that we used to have. If you don't have a stable career, don't get married and don't have kids!
On the con side, we have equally false and/or silly arguments. We hear that it will cause a loss of jobs. No study has ever supported this notion, and we have plenty of evidence from past raises in the minimum wage demonstrating the falsity of this notion.
We hear that most of the people making minimum wage are teenagers. Sudies have completely debunked this myth as well.
The people who oppose minimum wage laws are cut from the same stripe as those opposing child labor laws, workplace safety laws, workers' right to organize into unions, and on and on.