What should I do if my baby has nasal discharge?

Suction the nose. Use saline nasal drops and suction the nostrils. This can be done few times a day but not too frequently as it may irritate the nostrils.
Depends. If your baby has nasal discharge and he/she is not bothered by it and sleeping and eating fine, i would not do anything. If it's bothering them a little, you can suction their noses, use saline drops/sprays, elevate their heads, and use a humidifier or vaporizer. If it persists more then 2-3 weeks without improving, i would call your pediatrician since it could be a sinus infection or allergies.