Despite a large literature finding negative impacts of minimum wage increases on employment opportunities for low skilled workers, a few studies have found minimum effects. For a list of annotated references see: Do Minimum Wages Cause Unemployment).
While some of the studies finding no negative impacts use very rigorous experimental designs, they are still not perfect. There are a lot of margins that impact an employer's response to labor costs, and cutting jobs is only one among many of possible impacts. We can't always measure the relevant variables or outcomes that matter. Many of these impacts are in fact long term as well.
A very good post regarding this as well as the issue of survivorship bias can be found here:
"Many of the effects of increasing the legal minimum wage, for
example, take the form of actions that never occurred and hence cannot
be observed, for example, jobs that were never created because at the
higher minimum wage entrepreneurs did not consider the formation of
certain types of new firms or the creation of certain types of new jobs
to be worthwhile.
In short, in gaining a solid understanding of economic events, we
must beware of survivorship bias and never fail to consider the unseen
as well as the seen consequences of government interventions in the
market. A corollary is that we must not fool ourselves into the naïve
positivist belief that only countable data deserve consideration in
scientific work. The seen and the unseen, the counted and the
uncounted—all are proper raw materials for the serious and properly
trained student of economic and social life."
It is also interesting, in some of these states I have heard that there are some built in safety valves. For example, provisions that back off on the planned wage increases if detrimental effects become evident. It is interesting that politicians are aware that these kinds of wage increases are drastic and unprecedented, and these safeguards indicate they at least are cognizant of the evidence and plain common sense that indicates this could be bad. Actually, maybe that's a positive in terms of moving in the right direction of economic literacy? Unfortunately, this will only be based on short term and obvious measures like unemployment rates. Really bad outcomes like reductions in job training and job creation won't show up. The hidden costs won't get counted.
Some politicians have publicly stated that despite the economics, the arguments for these policies are morally compelling. But why not put their money where there mouth is if they truly believe that? If they truly believe that people are entitled to a minimum standard of living they could just as well legislate an income supplement or negative income tax equivalent to achieve this.
There is simply no way to argue that those that benefit from these policies 'gain' more than those that lose their jobs, or never get the opportunity to begin with because the new business that would otherwise employ them never opens as a result.
While some politicians want to make it illegal for the poor inner city kid to advance their career potential working for $7.50 an hour at McDonalds, it will remain perfectly legal for the rich kid in the suburbs to take the unpaid summer internship with the investment bank at $0/hr. That's how regressive progressive policies will ensure the poor get poorer and the rich get richer. But if you are in the class warfare business its a great way to generate repeat customers (voters) and garner support from the affluent progressive parents of those interns. Not a bad gig huh?