Often analytical (rather than emotional) minimum wage proponents admit a small decrease in total employment but think the benefits to those who make the cutoff helps offset that. That might even be true in a growing economy with generally low unemployment and short unemployment terms. In one with ~20% youth unemployment, and workers discouraged by multi-year stints of unemployment, any job – even a low-paying job – is better than prolonged idleness. Employers are understandably loathe to take a chance on those who have gone years without a reliable work history, or never started a work history at all.
When he wasn't quite as partisan, even Krugman could make an eloquent case that bad jobs at low wages were better than no jobs at all at wishful-thinking wages. See for example:
In Praise of Cheap Labor: Bad jobs at bad wages are better than no jobs at all
http://www.slate.com/articles/business/the_dismal_science/19...
Even today, while Krugman is against any sort of lowering-nominal-wages or lowering-minimum-wages for macroeconomic adjustment, he'd like to achieve the same thing through massive fiscal and monetary stimulus, essentially lowering wages through inflation.
In the funhouse mirror world of politics, the 1-2 punch of higher minimum wages but also inflation is a perennial winning strategy. Give people a superficially popular, though actually harmful, boost in minimum wages: an easy focal point on which to campaign. But prevent its full damage with harder-to-assess, noticeable-only-after-the-election inflation. Or at least keep the celebrated raises below other growth/income indicators, so that it's mostly a symbolic measure. (Any raise large enough to make a 'big difference' for minimum-wage earners would also make a 'big difference' in boosting unemployment.) Via the only-slightly-harmful symbolism, everyone (who is a politician) wins!