There is a condition called dacrostenosis, which is a narrowing or blockage of the tear duct that runs from the inner corner of the eye down into the nasal passages, draining the tears away. In young babies, it can be normal for this duct not to be totally developed yet. This leads to leaking of tears from the corner of the eye. Usually it clears up by around 1 year of age as the duct grows and spontaneously opens.
Therefore, in young babies it is usually best just to wait and see what happens. The condition is not typically dangerous, although sometimes there can be a collection of fluid or pus inside the partially blocked duct. In order to prevent this, you can "milk" the duct daily up into the eye to express any material there and to keep the area clean.
If your child is growing older, then there is less of a chance that the condition will clear up on its own. In this case, you may need to see an ear, nose, and throat surgeon
about opening up the duct. Sometimes this can be done simply by probing down into the duct, although other times a larger surgery