It seems reasonable that the minimum legal wage should not equal the prevailing wage for jobs requiring advanced training. I doubt (with only shallow research) that jobs requiring degrees pay in the $15/hour range. Through a vague hip-shoot google search, I found a source that says that 2012s graduates have a mean salary of about $45K - somewhere in the $23/hour range. Of course, that's for a 4 year degree - but graduate from a 2 year school with a certificate/degree in a technical skill often make more intiially than their 4 year counterparts.
All that being said, I still don't have an opinion on what the minimum wage should be. I'd prefer to hear what the minimum wage should accomplish first.
Someone else mentioned it, but check on a state-by-state basis. It's not easy to find that data per state. THe raw data is available, but it's generally separated by job categories, and trying to go through a few thousand options and pick out the degree-requiring form non is more time than I want to invest
But for a personal example, I attended the major state two-year college in Arkansas. My salary, upon finding a job after graduation, was about $36k. At that, I was the highest paid graduate they'd had in quite some time (on my side, I had two years experience in a work-study program working in my field, as well as years of military experience under my belt). The average 4-year salary here is probably about $40k.
The state-by-state data is important, because the states with lower average incomes will be affected the most radically by something like this.
And as I said, the $15 an hour wage is probably close to the high end for a two-year degree here (perhaps closer to average elsewhere), with $20 an hour being the low end for a 4 year degree (which seems reasonable with a $23 per hour average).
But I definitely agree with your final paragraph. I don't know what it should be, either...but $15 seems a bit high to me. There are a lot of "ripple effects" from increasing minimum wage; a lot of people seem to think it just magically makes everyone able to afford more. While it does increase buying power for lower-end jobs to some degree, it can also inadvertently tighten the job market (as companies hire fewer employees to cut costs), reduce incentive for college education (if I can make $15 an hour without, and $23 an hour by going $30k in debt for a four year degree, is it really worth it?), and have all sorts of other effects that rarely seem to be accounted for.
I want better data, myself
However I feel this ripple effect could cause real problems for already struggling industries.