Breastmilk is good for everything
Seriously, try cleaning the eye from the corner out, with a warm washcloth, then drip breastmilk in it and let it dry. I'd probably repeat every 2 or 3 hours--or as often as the baby nurses. A blocked tear duct isn't the same as conjunctivitis, but basically the breastmilk absolutely won't hurt--and could do a lot of good.
Just in case it is
conjunctivitis, be sure to wash your hands a lot, never reuse a washcloth or towel for the baby, change bedding frequently, and keep the other kids clear of it. It's not fatal, but it's annoying and highly contagious. The first time Clara picked up pink eye--full blown, and soon followed by mine and my husband's bout--the ped said that if left untreated, it will generally clear on its own in a week or so. We treated with antibiotics only because she's in daycare, and would not be allowed back until free of symptoms.