To treat a blocked tear duct, doctors usually recommend massaging the tear ducts several times per day along with applying warm compresses. To safely massage the tear duct, wash your hands, then place your finger along the side of your baby's nose. Apply slight pressure while pressing down toward the corner of the nose. Applying warm compresses to the eye will help promote drainage. (Because infants could easily be injured if not performed properly, ask your pediatrician or eye doctor to demonstrate the massage technique.)
Pediatric doctors sometimes prescribe gentle antibiotic eyedrops or antibiotic ointments to resolve any possibility of conjunctivitis or other infection. Most doctors are conservative in their treatment, as most cases resolve completely on their own. It is reassuring to know that 95% of these conditions go away without treatment within the first year of life. If the condition does not go away, your doctor may probe and dilate the tear duct. If necessary, a tube may be placed in the tear duct to ensure that it stays open. Probing is usually not performed until at least six months of age. These procedures are usually performed under general anesthesia.
Source: Rosenbloom, Alfred A., Jr. and Meredith W. Morgan. "Principles and Practice of Pediatric Optometry." ISBN 0-397-50917-0, Copyright 1990.