A photo of the mouth at rest and one smiling would GREATLY ease our job and increase the accuracy and relevancy of the answers.
Besides the NATURAL facial asymmetry we all have, there are several reasons why one side of the mouth and "THE" solution would greatly depend on the reason for the droop.
Age related drooping associated with a deepening nose to mouth (Nasolabial) fold would be improved with a mid-facelift as well as with fillers in the fold.
Sometimes the drooping is seen on the NORMAL side because the nerve supplying the muscle pulling the corner of the mouth down (DEPRESSOR ANGULI ORIS) , has been damaged. (Such a birth injury related to a forceps injury (A famous example is Sylvester Stallone), the same injury can also happen during a neck lift when a branch to that muscle is damaged). Such drooping can be made symmetrical and nicely improved by weakening the muscle which pulls the corner of the mouth with a few units of Botox.
Peter A Aldea, MD