How to fix mouth corner asymmetry? (photos)

I believe you could see very good results by injecting small amounts of filler into the corners of the mouth. This treatment does require precision in both placement and dosing amounts to get it just right. If performed correctly, it will effectively address the downturned corners as well as the minor asymmetry. Find a board-certified dermatologist with extensive experience and training in fillers to ensure an optimal outcome. 

fillers, threads, botox and morpheus can all help. based on your pictures though you have alot of loose skin and likely would benefit from sculptra and/or accutite or early face/neck lifting.  hormonal optimziation would help as well to limit early sagging (see link).For facial rejuvenation, 
1) skin care with products and PRP/microneedling
2) PDT light therapy few times a month with growth factors
3) laser treatments like clear and brilliant regularly and a few times a year Fraxel/halo Viva, Co2/Erbium
4) Sculptra injections a few times a year for collagen production with fillers like juvederm/restylane/belotero/radiesse/bellafill
5) thermage/ultherapy for yearly skin tightening with or without PDO or cone based instalift threads for lifting
6) botox every 2-3 months to stop movement lines and prevent lines at rest
7) deep microneedling RF like Fractora/morpheus8 for tightening of the skin
8) Vascular laser like vbeam/excel V or BBL forever young treatment every quarter (3-4 months)
Best, Dr. Emer.

Dear whowouldcare,
I perform a lot of lip injections with fillers such as juvederm and it works great. Many times I need to put more in one side than the other to attempt to even out the vermillion border (The red line of the lip). This helps provide more symmetry at times.For patients looking for a more long lasting result, I recommend fat grafting. That is, using your own bodies fat to make your lips naturally look more shapely. The bonus is that the results last many many years.
In general, I avoid using permanent prosthesis because I find that the lips are such dynamic and sensitive structures, the end result is not natural.
Daniel Barrett, MD
Certified, American Board of Plastic Surgery
Member, American Society of Plastic Surgery
Member, American Society of Aesthetic Plastic Surgery

A small amount of filler placed by an experienced injector will provide some support to the lower corner of mouth. Botox or Dysport injection into the DAO muscle may help but I usually recommend starting with the filler first.