Let's start with the benefits of eucalyptus and what you should be doing. First of all, little koala, it smells lovely. Eucalyptus is a cool scent along the lines of mint and rosemary, and it's often used in the same products. The scent tends to relax us and is sometimes found in calming products like the De-Stress Spray from Babylonian Soap Company.
Folk medicine has long mentioned anecdotal stories of eucalyptus being used to clear your breathing tubes. Please note that this is through scent; again, it's anecdotal. Still, brands like Williams Sonoma offer a rosemary eucalyptus spray to put on your shower walls and inhale. You'll also see it used as an ingredient in products like Vicks VapoRub.
Eucalyptus is often used in bug repellents like Repel and Off!, and is also the main ingredient in many products to help inflammation. There was even a 2013 study published by the National Center for Biotechnology Information shows that inhaling eucalyptus oil made participants feel less pain. There is also some evidence that it lowered blood pressure. Please, though, if you're having medical issues, do see a doctor, and don't rely on a pretty shower accessory as your sole M.D.