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

You would get a nice improvement with lip fillers. These fillers, (Volbella, Restylane, Juvederm) are good to create a symmetrical lip. Nova threads can be used to improve lift and shape. Implants or fat are alternatives. Technique will be important here, so you'll want to see someone meticulous. I recommend getting a formal evaluation with a facial sculpting expert. Best, Dr. Emer

An advanced Botox® injecting technique involves injecting microdoses of Botox® Cosmetic equally into the sides of the chin (or preferentially on one side over the other to correct asymmetries), and prevent a downturn of the mouth comes. As this is a procedure requiring expertise, please consult a board certified aesthetic dermatologist for a consultation.  

Thank you for your question! These fillers, (Volbella, Restylane, Juvederm) are good to create a symmetrical lip. Nova threads can be used to improve lift and shape. Implants or fat are alternatives. Technique will be important here, so you'll want to see someone meticulous. I recommend getting a formal evaluation with a facial sculpting expert. Best of luck! 

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.

Thank you for the question and photo.
Unfortunately it is very difficult to give you advice without seeing how your mouth actually moves. Small amounts of toxin (such as BOTOX) can be used to correct facial asymmetries such as this and are commonly performed in cases of mild facial palsy or congenital asymmetry.
I would suggest arranging an in person consultation with an experienced facial plastic surgeon. 
Good luck!
Dr Christopher Macdonald
CRM Aesthetics 

I'm not sure that one side is going down, I think the other side is going up.  You should find something else to worry about because 1) there is nothing wrong with you and 2) there is no way to remedy this.  It would be real easy to convert something that only you can see into something everyone can see.