Economist David Neumark has a nice piece in the WSJ today regarding the upcoming increase in the minimum wage. David (a democrat as far as I know, though it should not matter) has done a long series of valuable papers (and apparently now a book) on both the minimum wage and its awkward cousin, the so-called "living wage".
The literature on the minimum wage is far from perfect, largely because the available variation is not perfect. Using variation in the national minimum is pretty hopeless as its effects get swamped in the time series by the business cycle and many other policies. Using variation in state minimum wages is complicated by the fact that the states that raise their minimums are not a random sample of states and that the timing of state level minimum wages increases tends to coincide with booms, because the economic, and thus political costs, are lower then (though clever Michigan bucked this pattern with its recent increase). Further complicating things is that many of the likely effects of a minimum wage do not appear immediately, but rather over time as employers substitute capital or foreign labor for domestic low skill workers (as when you fill your own drink at McDonald's or Chipotle). Expectations also play a role here, so that minimum wage increases can have not only lagged effects but also anticipatory effects. It all adds up to a pretty rough road for empirical research, which is presumably part of the reason for some of the conflicting findings in the literature.
My read is that minimum wages persist because of an unhappy coincidence of public ignorance of basic economics (in this case, as in so many, a failure to see the indirect effects of a policy along with its direct effect as well as ignorace of its poor targeting), the interests of unionized workers, who want to raise the relative price of their non-unionized competition, and the interests of high wage / high price states that want to raise the relative price of labor in low wage / low price states.
This is yet one more case of evidence-based policy; in this case the policy is the opposite of what the evidence suggests it should be. It is also another opportunity for some change one might believe in. I'm not holding my breath, despite all the fine economists in the administration.