Nasal discharge is NOT good in a duck. This can mean a respiratory infection or salmonella. I own a duck farm, and runny noses always get the red alarm.
Sometimes there will be water runoff after drinking, when a duckling isn't swallowing all the way. If the discharge is only after drinking, then your duck is just dribbling.
If it's not just water... Do NOT touch the discharge. It likely is shedding bacteria, and you can get infected.
You can take your duck to a vet at this point. Here at the farm, we go get the antibiotics from the farm supply store. But, experience means that we can pretty accurately determine what's wrong with the duck and which antibiotic it needs.
The first step to treating a sick duck is to move it onto a mesh screen which will allow the poo to drop below the duck. This keeps the duck from reinfecting itself. Wash everything the duck touches in alcohol or vet grade disinfectant (which you can get at a farm supply store). Stall wash works well for this, in the horse section.
Once you've quarantined the duck, determine what's wrong with it. Is it wheezing, breathing with an open beak, or sneezing? Repiratory infection.
Although, honestly, if it's just a runny nose and sneezing, sometimes ducks have allergies to certain molds and pollen. I've had ducks allergic to goldenrod. So, see if changing his location helps at all before you panic.
Is it having watery green poo, loss of balance (only in young ducks, not older than four weeks), nasal or eye discharge, lethargic? Salmonella. This is a digestive infection, but the bacteria sheds from every part of the duck. Avoid touching the duck, and wash your hands every time you handle it (have hand sanitizer ready).
You'll need to either take it to the vet, or get an antibiotic from a farm supply. If you get your own antibiotic, follow the directions to the letter, or you'll end up with a sick duck and an antibiotic resistant infection.