How to remove silicone caulk from a shower wall?
If you saturate toilet paper with the remover and slap it on your vertical surface it will stick there and hold it against the caulking. Practice a couple times with water to get a feel for the appropriate amount of moisture. Scrape off as much as possible before you start in with the remover though.  Just to clarify, you should be using a plastic blade in the scraper so you don't damage or leave marks on the wall.

EDIT: BTW, you can also use acetone (i.e. nail polish remover) to remove silicone caulking. Acetone attacks some plastics, so I'd only go this route on tile.

EDIT 2: If neither of those options work, you can also go to an auto supply store and pick up an adhesive eraser wheel. You use these with an electric drill and they are designed to take adhesives off of auto bodies without damaging the paint.  Again, this might not be the best thing to use on a plastic surround.