The thing I've been fearing would happen just happened: I fell down the stairs while wearing the Baby Bjorn. (Actually, my worst fear was falling down the stairs while carrying Austen in my arms, but this was almost as scary.) Luckily I only fell down the last two, and I seem to have instinctively twisted onto my side to protect Austen, who was unhurt. He seemed startled and puzzled, but he didn't cry, even when it took me a few minutes to get back up.
I banged my left knee and my right elbow pretty hard, and I seem to have twisted my back and my right ankle, but I'm still mobile (and still wearing the Bjorn), so the injuries must be very minor. Next time I will make sure that I really am completely down the stairs before I start reading the Wall Street Journal.