Some research told me these were spider mites. And if left unchecked they will kill the plants.
More research told me that neem oil sprayed on the plants would disrupt the mites' reproductive cycle. But I would have to do this regularly until they were gone. I wanted to go this route instead of right into pesticides because of the cat. He loves the balcony and loves sticking his face into all the plants he can reach. I did not want to risk any harm to him. So I got some neem oil, got myself a pressurized mister, and got to work.
I sprayed the whole garden down several times. Unfortunately we were away a few times in between sprays so it wasn't done as frequently as ideal. The mites primarily live on the undersides of the leaves, therefore the spraying must occur on the undersides of the leaves too. This is a pain in the butt and I went through A LOT of water+neem oil on each spray. In addition to spraying the plants, I had to contain the water so it didn't leak down onto my neighbour's balcony. I think what I achieved, if anything, was to hold the population in check...except on the lettuces. They just ate those up and I had to get rid of all my lettuces. I gave up after several sprays. The amount of work that went into spraying the garden wasn't worth the effort for me.
I got rid of the heavily infested plants (the delicate, delicious ones) and kept the hardier, but still infested, ones. Unfortunately, as fall is coming on I will have to get rid of them all. Even our precious Pisano (the palm). He's a big, strong plant, but he's thoroughly infested.
Indoors I have two large plants that are strictly houseplants (Domingo and Fernandina). Both are grown from babies my mother gave me, and their mothers continue to grow large and tall in my mother's house. I love these two, and will not risk their well being. Therefore all the plants on the balcony will be bagged off and carted out of here with nary a mite escaping.
So long balcony garden. We'll see what new hope spring of 2019 brings.