Deck Staining with a little light drizzle
It depends. 

Let's back up for a second and talk about how stain/waterproofing works. Wood is a porous material. When dry, it will soak up liquids like a sponge. In fact, it's that soaking/drying cycle of water that will eventually wreck your wood (and why waterproofing helps extend the life of exterior wood). Deck stains generally rely on that principle. The wood soaks the stain (or waterproofing) and thus it penetrates into the top layers of wood (latex relies a bit less on this, and tends to coat the outside a bit more).

If your wood is wet, even a little, it interferes with this because the wood has absorbed some water, and thus the pores that would fill with stain are filled with water instead. If your stain is oil-based, it just won't penetrate as well. If it's latex, the water will wreck your finish because latex is designed to break up in water.

The real question isn't about drizzle, it's about how wet your wood is. If we're talking about one of those wintery "fine mist" drizzles, the wood might stay relatively dry enough you can still apply stain. If we're talking about anything heavier, I would say probably not. 

Either way, I wouldn't risk it. Just wait for a dry day. Eliminates any question of how well the stain will work.