Dear friend. You can apply some artificial tears to moisten the baby's eye. If your baby doesn't sleep well in the night, this may cause its red eye. But don't worry. This will disappear very soon because this is just lack of sleep, if it catches up enough sleep the next day, the symptom will go over on its own.
I think you should take your baby to see an eye doctor first. We should apply prescribed eye drops for babies. If it is a sign of the beginning of an eye infection, to see an eye doctor immediately can prevent vision problems from happening. Baby can't tell their feelings. So we can only ask the doctors to diagnosis the problems for them.