First I'll talk about the empirics, then some theory, then some thoughts on the discourse around the issue.
Bob Murphy and Ryan Murphy (no relation) both give us a look at the state level data, comparing states that have minimum wages higher than the federal minimum wage with those that don't. David Henderson also favorably shares a comment from Krugman's comment section that talks about a time series of teenage employment from Casey Mulligan as "empirical evidence that the last increase in the federal minimum wage... really did increase unemployment among the least skilled" such that Krugman needs "magical progressive fairy dust" to justify his position. Now I know Bob, Ryan, David, and Casey understand the problems and limits of these sorts of approaches, but this all seems to give the impression to people that this gives us our answer - so it's worth pointing out why methods used by guys like Card and Krueger are celebrated and widely appealed to in issues far afield from the minimum wage (for example, even guys like Don Boudreaux and Russ Roberts favorably cite people like Valerie Ramey and Robert Barro when they use very comparable methods to get at fiscal multipliers).
Minimum wages, like most things in economics, are not randomly assigned. Time periods with higher minimum wages are different from those with lower minimum wages and states with higher minimum wages are different from states with lower minimum wages. So the trick is to find an exogenous (essentially, a random) change in the minimum wage and to clean out the other correlates of minimum wage levels.
Bob and Ryan's approach doesn't help here. Yes, California and Mississippi have different minimum wages, but there are a lot of other things different about them that a simple comparison is going to pick up. These other things may be causing the minimum wage, or they may be completely unrelated to the minimum wage, but they could still be picked up in the minimum wage variable.
The most natural solution has two parts to it. The first is not to look at minimum wage levels, but to look at changes in the minimum wage. Presumably these are sufficiently discrete that they are not caused by changes in ideology, price levels, demographics, etc. Second, you'd like to do this exercise for very similar local economies - one that changed the minimum wage, and one that didn't. This helps you to identify a counterfactual using the state that did not change its minimum wage, so you compare the changes in the state that did change to the state that did not. This approach seems to be our best bet for getting a clean estimate of the effect of the minimum wage, and this was exactly Card and Krueger's (1994) approach. That paper isn't the end of the discussion, but it is rightly considered a classic. People reading critical libertarian blog posts on the issue should not be confused about the fact that this is an important paper and rightly respected paper.
If you were convinced by Bob Murphy's data work, just ask yourself - (1.) didn't a lot of those minimum wage states also have big housing busts? (2.) don't they have a lot of other progressive policies or regulations that might be driving this? (3.) hasn't there been an oil boom in some of the lowest teen unemployment states? (4.) don't these teen unemployment rates correlate well with general unemployment rates and might that be caused by some other common cause? (5.) don't the lowest teen unemployment states on his list have a lot fewer minorities?
The Card and Krueger study isn't perfect, but it doesn't have any of these problems which in my opinion make Bob and Ryan's approaches non-starters.
You may be hearing a lot about a more recent study by Dube, Lester, and Reich (2010). The point of this paper is to generalize the Card and Krueger approach to differences across many more borders than the Pennsylvania/New Jersey border: