Both are right, in a way. The Republican argument that a $15 minimum wage would eliminate all the jobs currently getting paid less than that is simplistic, ignoring the fact that when you put more money into low-income people's pockets, they spend it and stimulate the economy. And it seems harmless to take that money out of the pockets of large corporations that, for want of good investment opportunities, have been just sitting on it for eight years. But I think it's also dangerous to completely discount the job-killing potential.
Let's start with a reductio ad absurdum: if a $15 minimum wage is good, why not $20, or $30, or $50, or $100? At any given minimum wage level, workers whose value to their employer is less than that will be unemployable. With a $100/hour minimum wage, either (a) the majority of working Americans lose their jobs, or (b) inflation devalues the $100/hour until it matches their job productivity. Now, a certain amount of inflation would be welcome in the current economy (by everybody without a large bank account), but betting that the inflation would kick in before the unemployment did sounds risky to me. Anyway, it seems clear that above some level, a minimum wage will cost jobs; the question is whether we're currently above or below that level.
This is now a quantitative, empirical question, not a qualitative matter of principle, and its answer probably depends largely on geography and the local cost of living. $15 in New York City is a lot less money than $15 in rural Iowa, which is in turn a lot less money than $15 in Puerto Rico. A $15 minimum wage in New York City would be almost adequate to live on, and stimulate the local economy; the same wage in Puerto Rico would be comfortable for the people who got it, and could well devastate an already-struggling local economy by making many people unemployable.
So what would be a better solution? We can reasonably consider both a government solution and a free-market solution. The government solution is to set minimum wages based on the local cost of living (although finding the right formula to appropriately weight local factors against non-local factors would be tricky). (I guess another option would be setting minimum wages at the state level, as at present, but that predictably produces a "race to the bottom" effect.) The free-market solution is for workers and their employers to negotiate geographically appropriate wages. But we all know that when an individual, non-superstar worker "negotiates" with a multi-national corporation, the latter is in a much stronger bargaining position and will probably pocket almost all of the worker's productivity. The obvious counterbalance to this is unionization: if we still had strong unions in this country, we might not need a minimum wage law at all.
In short: if you don't want to see a nationwide minimum wage law, encourage your workers to unionize instead.