I put Starsan in a spray bottle and spray the hell out of my wine theif about 5-10 mins prior to taking a reading. Just make sure you get starsan/water mix inside as well since it will be touching the beer.
I use a wallpaper tray I bought at the hardware store. $5 and I can keep the thing submerged as long as I like.
If you're spraying with Starsan, just make sure you keep the surfaces wet for the required contact time and you'll be fine.