Most people who argue against the minimum wage assume labor demand is elastic, meaning the amount demanded will disproportionately change relative to price changes. Here's a simply supply and demand chart to illustrate:
In the above chart, the government increases wages from P to P1 and demand drops from Q1-Q2. In this example, the line Q1-Q2 is longer than P-P1, meaning the increase in price had a large impact on demand.
However, what if demand is inelastic? In that case, we'd be looking at the following chart:
Above we see the quantity demanded decrease slightly relative to the larger increase in price. This means that even if wages increase, we still demand a certain amount of labor.
The second chart is actually far more realistic. Consider the following fact pattern: a store currently has 100 employees. Because they're a profit maximizing entity, they are already using the optimal amount of labor (give or take say 5-7 employees). Let's assume they cut their workforce by 25% because of the increased wages they must now charge. At this level of cuts they'll see shelves stocked less frequently, fewer employees helping customers finding items and longer check-out lines. This will actually decrease their sales; customers will get tired of not being helped and will become frustrated with the longer lines.
The underlying reasoning is based on the production function, another basic micro-econ concept:
Anyway, food for thought.
Thanks to Jared Bernstein for first pointing this out awhile ago.
[UPDATE]: Alan Kreuger's research confirms.