I asked friends online, and one recommendation was to use a splash guard. The problem is that these need a right angle between the wall and the tub, and our house has few right angles. Instead I followed a different suggestion and installed a piece of waterproof (plastic) shoe molding with silicone caulk:

This looks a little silly, but works perfectly. No more water on the floor!

(One concern people had was that whatever barrier we put here would collect stagnant water and get moldy. It turns out that enough water flows along this that it gets flushed out pretty consistently, so not a problem.)