If his eyes are very red with copious amounts of pus coming from them and he is less than a month old you should seek medical advice urgently.
The majority of babies however just have a blocked duct which does not require antibiotics. Your midwife or health visitor should be able to show you how to clean your baby’s eyes.
The naso-lacrimal duct usually canalises around about 36 weeks gestation but 20% of newborns show signs that one or both are blocked. Eyes start watering within 6 weeks of birth, they get sticky eyes and occasional conjunctivitis which is when the whites of the eyes go red. We would usually take a swab and treat with antibiotic eye ointments only if the eyes are red.
You should massage the sac between the corner of the eye and the nose (see diagram above) 6 times per day and clean with cool boiled water. 90 – 96% of cases resolve by one year of age. Your GP can refer your baby to a paediatric ophthalmologist if he is one of the 4% who still has sticky eyes at a year of age.
There is a good parent information leaflet on this topic here.