How do I shoot the propellers off of the Guardian Skywatcher?

Just aim at the propeller "motor" bit. They don't take much damage from regular arrows, but two or three bomb arrows can destroy a propeller pretty reliably. I haven't experimented with other arrow types yet.

Here's a video: IGN

I've found using an ancient arrow at the propeller motor will destroy it outright. Note that it can still keep flying with one motor left, so you need to take down all 3. I haven't really experimented with other arrow types. Using a bow like the Korok bow which shoots 3 arrows simultaneously will instantly kill a flying guardian if all 3 hit.