With natural hair brushes I tend to hang them, mostly because of habit and not wanting a funky smell. With synthetics I tend to stand them, but mostly because I don't have enough stands.

I don't know how much it really matters providing you have done a good job cleaning the brush after use.

I guess the truth is that hanging a brush upside-down will not harm it, but leaving a wet brush to stand might...

There are other factors like air flow in the room, humidity and so on but let's not open that can of worms.