No, goopy eyes will not contagious except such symptom is a sign of pink eye. By the way, my mum said the breast milk has curative properties due to its antiseptic property, particularly when you encounter with goopy eyes. As for precautions, you are advised to avoid wiping off the substance around the eyes by hands directly, instead a swab will be appropriate. Viral pink eye is a contagious form. To some sensitive individual, the only way to prevent is to compeletely avoid contacts with patients. However, if you just get a little dirt or something in your eyes, you just need some boiled water(have been cooled) to wash your eyes and wait a while to see if it gets any better. Well, If pink eye outbreaks near you recently, you should take this possibility into consideration, especially to the newborns with immature tear duct. It is wise to go to your doctor, take some medicated eye drops after a simple examination. Besides, you can place warm tea bags on your eyes until you see the doctor. Just for reference.