I have seen a drain, that is equipped with a small, (I think it was a ¼â€) water supply line that kept the drain primed. I don’t know how you install one of these drains, (with out taking out a lot of drywall on the first floor ceiling) and then you would have to tap into a water supply line.
I don’t know if they make a retro fit device but, here’s a hair brain idea. You could remove the existing cold water faucet and install one that has that side port, (the type that you can branch off for a swamp cooler) and run a ¼†copper tubing to the drain, and then at the end of the tubing install a pressure compensation, (very low flow) irrigation type drip head. Along with this installation, you could even install a in-line ¼†valve, (up by the faucet just after where you installed the copper tubing) where you could regulate the flow to the drip head.
A simpler fix would be to poke a small hole in the bottom of the drain line to the washer, and every time the washer drains, a small amount of water would hit the floor. A draw back would be is that you would probably have some soap scum build up on the floor, by doing it this way.
Like I said, these are just a few hair brained ideas.
Good luck,
Mike