Unfortunately, the entire Basset Hound breed is prone to a condition called Cherry Eye. It's called this due to the appearance as seen in the photo to the right. It happens when their third eyelid pops out, creating an irritation.
Please note, this is not the same as simply seeing redness due to their naturally drooping eye lids. Cherry Eye looks like a bubble protruding out of the eye area.
Occasionally prescription steroids or ointments can strengthen the eyelid enough to go back down by itself. If that doesn't work, (which is often) this is a fairly minor and frequent occurrence in Basset Hounds, and is often treated by vets as a simple surgical procedure. Recovery is quick, being sent home with pain reliever, ointment, and possibly antibiotics. Then they're all better; hopefully for good. :)
If left untreated, this condition has the potential to turn into a more serious problem, so it is best to get this corrected promptly.