Can I change one side of my mouth going down whenever I talk or smile? (Photo)

This is a minor asymmetry and I am inclined to tell you to just leave it alone. If you insist though, a small amount of Botox or Dysport could be injected into the depressor anguli oris muscle (DAO) to raise the corner of the mouth that is down. 

You likely would benefit from injection of a small amount of Botox or other neuromodulator into the Depressor Anguli Oris (DAO) muscle to reduce this movement. This needs to be very carefully placed so as not to alter the smile adversely, so ensure you seek help from a highly trained and experienced physician injector such as a dermatologist.