What is causing this water on the drywall and how concerned should I be?
What room is this in? 
Is the room near the kitchen or the bathroom? 
Is that an exterior wall?

I will see this in our bathroom when I forget to turn on the fan, or when I run the humidifier in a room with a cold exterior wall.  

It's likely just condensation from cooking, a humidifier, the shower, etc. 
The drops are darker because as they roll down the wall they pick up dirt, dust, and grime (You never notice how dirty your walls really get).