There are many causes that can cause discharge in the eyes and runny nose. And the most common reason is cold with hay fever. Aside from eye discharge and runny nose, cold may also cause headache and sore throat. Although cold is the most common diseases for human being, you shall treat it serious especially for babies. Or it may develop some serious problems. Anyway, you shall be careful to apply medicine for baby. So, consult an eye doctor first.
It is not easy to tell what cause the discharges and runny nose for your baby. Usually, pink eyes and eye infection are blamed to the excessive eye discharges. But I seldom heard that people with pink eyes or eye infection will get runny nose. It seems much more like a hay fever. It causes by the allergic reaction to the outdoor pollen. And the hay fever is possible to cause nasal congestion, thus make people get runny nose and some eye problems such as itch or burning eyes, also, discharges sometimes.