I asked at The Barn, and they said that basil is notoriously challenging to start this time of year. It wants warm soil. In fact, they don't usually even try to get starts for a couple of months.
I ended up double stacking the basil starts on top of a whole bunch of smaller pots with starts in them: tomatillas, tomatoes, chrysanthemums, peppers, and yes, more basil underneath.
Two days later, I had chrysanthemums sprouting. The next day, the basil was sprouting, and more come through the soil everyday.
We'll see how that goes.