It is possible that the swollen and red eye of your 6-month-old baby could be conjunctivitis. Conjunctivitis often resolves on its own within 24-48 hours, but if the symptoms persist or worsen, it would be advisable to consult a healthcare professional for further evaluation. Cold compresses can help reduce inflammation in the eye. If the eye is stuck shut in the morning, continues to discharge throughout the day, or remains very red for more than 24-48 hours, it may be necessary to consider antibiotic treatment. If unsure, a pharmacist can provide a preliminary evaluation of your child and offer guidance as needed.