The photos are not quite in focus so I won't swear to anything, but this image, below, just looks like root primordia. Basically, they are baby adventitious root nubs.
Most gardeners are probably familiar with seeing root primordia on tomato stems but most other plants, like basil, can also develop it. If the environment of the Aerogarden has really high humidity, that could be tricking the plant into thinking that it should develop roots there.
The combined symptoms of root primordia, browning lower stems, and cell rupture at the base of the stem makes me think that your plant may be absorbing way more water than it can tolerate. The humid conditions and split stem also make a perfect environment for microbial pathogens to take hold. Keep an eye on the browning and watch to see if it starts spreading up the stem.
Can you lift up the little grow basket and see what the roots look like and take a photo?