Eye infections are very common in kittens. Usually they are not serious.
Because their immune systems are immature, kittens are highly susceptible to a number of viruses and bacteria that can cause swollen, weepy eyes. The most common offenders, herpes and Chlamydia, sound like venereal diseases. They aren’t. They are ubiquitous infectious agents, and almost every cat on earth contracts one or both of these germs at some point. Most are exposed as kittens.
The weepy eyes may take several weeks to clear up. A veterinarian should assess the kittens to make sure that nothing more serious is going on.
In rare cases, severe eye infections in kittens can lead to permanent damage and long-term eye and sinus issues. However, this is not common at all.
Therefore, the chances are very good that the kitten you are about to adopt does not have a major problem.