Does Botox help with the flaring of the nostrils when smiling?

Botox cannot change the width of the nose.  It can be used to treat lines and wrinkles associated with "bunny lines" (lines across the bridge) and the nasal flare (lines on the top side of the nose made with certain facial movements).  I would recommend that you go to experienced cosmetic surgeon to see if you would benefit from Botox.

Botox has an effect on muscle movement and can be used to diminish the movement of facial muscles. If the effects that concern you  (nasal flaring) are the result of muscle movement, then Botox will help. Botox alone cannot change the width of your nose or the dimensions of your nostrils. 

Dear ak2628,Botox is generally an on off switch for muscle activity. It will affect any muscle which is injected.  Finding an experienced physician injector is your best bet. Most of patients see first changes after 4-7 days and the final results after 10-14 days. It depends on the amount of Botox and area where it is injected but it usually last for 3-4 months.Daniel Barrett, MD, MHA, MS 
Certified, American Board of Plastic Surgery  
Member, Am. Society of Plastic Surgery

Botox has been used to decrease nasal flare for a long time. Very frequently injection requires much higher doses than 3 units. If you had no response to 3 units, one could inject at least another 3 units 2 weeks later. Eventually your flare will diminish assuming injections are made in correct area. Good luck!