Yes. Bryan Caplan has the answers:
1. Cutting wages increases the quantity of labor demanded. If labor demand is elastic, total labor income rises as a result of wage cuts.
2. Even if labor demand is inelastic, moreover, wage cuts reduce labor income by raising employers' income. So unless employers are unusually likely to put cash under their mattresses, wage cuts still boost aggregate demand.
An even simpler way to explain it: Imagine every firm divided its existing payroll between a larger number of workers. How is that bad for aggregate demand – or anything but good for employment?
P.S. If you prefer specific facts to textbook arguments, see Scott Sumner's legendary Table 12.2 on wages and the Great Depression.
As Bryan titles his post: "Cutting the Minimum Wage Really is Good for Aggregate Demand." The actual arguments in Krugman's blog post concern an overall downward spiral in wages and prices, not minimum wage cuts at all. The chance that minimum wage cuts set off such a spiral is very, very small. Krugman's third paragraph makes perfect sense but the fourth paragraph and onwards is simply discussing a different topic.
I would add two points. On Bryan's #1, workers at the current minimum wage are unlikely to receive nominal wage cuts if the minimum wage were lowered, for the usual morale and efficiency wage and lock-in reasons. So the chance that total labor income rises is very high. Second, no I don't believe in an upward-sloping AD curve, but in any case multipliers from production increases plus wage bill increases are likely to be more potent than multipliers from aggregate demand increases alone.
Addendum: Will Wilkinson offers relevant comment.