The evidence is consistent with the theory

From Myth or measurement: What does the new minimum wage research say about minimum wages and job loss in the United States? by David Neumark and Peter Shirley.  

As an international economics undergrad and international finance grad student, I have been watching this research argument for more than four decades.  Long ago I reached two conclusions.  First - In theory and practice, minimum wage legislation reduces employment.  Second - Whatever conclusion you reach, there is voluminous research to support that position.

It is interesting to see Neumark and Shirley's overview.  From the Abstract.

The disagreement among studies on the employment effects of minimum wages in the United States is well known. Less well known, and more puzzling, is the absence of agreement on what the research literature says—that is, how economists summarize the body of evidence on the employment effects of minimum wages. Summaries range from “it is now well established that higher minimum wages do not reduce employment,” to “the evidence is very mixed with effects centered on zero so there is no basis for a strong conclusion one way or the other,” to “most evidence points to adverse employment effects.” We explore the question of what conclusions can be drawn from the literature, focusing on the evidence using subnational minimum wage variation within the United States that has dominated the research landscape since the early 1990s. To accomplish this, we assembled the entire set of published studies in this literature and identified the core estimates that support the conclusions from each study, in most cases relying on responses from the researchers who wrote these papers. Our key conclusions are as follows: (i) there is a clear preponderance of negative estimates in the literature; (ii) this evidence is stronger for teens and young adults and the less educated; (iii) the evidence from studies of directly affected workers points even more strongly to negative employment effects; and (iv) the evidence from studies of low-wage industries is less one-sided.

At Marginal Revolution, commenter JVM notes

JVM

The thing that's so insane about all discussions of minimum wage is that it's framed as either "there's no disemployment effect" (meaning it's great!) or "there's a disemployment effect" (meaning, hm maybe there's a tradeoff).

In reality there is an adjustment happening whether it's disemployment or something else. The adjustment is the size of the distortion between minimum wage and market wage. Is the adjustment coming from: a) nonmonetary compensation? b) consumer prices? c) disemployment? or d) profits? Probably all four, probably long term and short term are not the same. Because capital seeks global average returns profits may be shocked short term but are probably not much impacted in the long term. If the long term adjustment is mostly a/b/c then it's not a great policy. The monopsony story ends up looking weak even if you believe in monopsony.

I agree.  There is a shock which temporarily raises costs.  How does the system respond?  There are JVM's four a) reduction in nonmonetary compensation b) increase consumer prices c) reduced employment or d) reduction in profits.  I would add at least one other system response which would be e) technology substitution for labor.  Sort of a specific aspect of reduced employment, but important because it is common.  I would also add f) labor transference.  Getting customers to do something that was provided by a worker in the past.  

So when anyone argues for a minimum wage that is markedly above the market, the question has to be, given the mix of industries and sectors affected by the proposed minimum wage, what are the likely responses across those industries and sectors in terms of lost nonmonetary compensation; increased consumer prices; reduced employment; reduced profits (and tax receipts); increased technology/labor substitution; and increased labor transference.

The reality is that no one can model those impacts across multiple industries and sectors so the intervention is a heartfelt and well intentioned stab in the dark with no confidence as to who benefits to what degree in what ways.

But you can be pretty confidant that in the longer run, higher labor costs will reduce demand for labor.