Drops

One cause can be a blocked tear duct. Suggest you consult your physician. 

I was told to put my middle finger over the inner corner of my eye after putting drop in with eyes closed obviously. Hold that pressure on for at least 10 seconds per drop. You could always use your thumb on the outer part of eye at the same time to ensure no leaks occur. You may have already tried this but it worked for me. Hope this helps

It is unavoidable even for doctors as some bottles are made that way. Simply dry with a clean tissue.

Be sure to use Punctal Occlusion so that the drop stays in your eye and doesn't go into the rest of your body.   Check out below link to put in your drops correctly.  I was doing it incorrectly for a long while.  And apparently you only need a micro drop

https://www.brightfocus.org/glaucoma/article/10-tips-using-glaucoma-eye-drops

Do not appy any product on  your eyes without doctor consultancy.

I have to do eye drops for the glaucoma and dry eye. If a little drips out it should be ok, always happens, just make sure you pull your lower part down a little, and if you feel you're missing completely, which I have, do it again. After doing both eyes, tilt your head back and press your pointer fingers close to the eye duct on the bridge of the nose, they say that makes sure some goes where it's supposed to, instead of your sinuses. But some always runs down my face. Good luck.