I slide the entire strainer assembly into the fridge. It keeps draining as I use the yogurt, so there are never any nasty puddles of whey. I hate those. And since I've gone this far without washing much of anything, why start now?
But there is one thing: I want to save some of each batch to inoculate the next. So the last few ounces out of the jug go into a jar. I leave this starter untouched for the week in the fridge.