There a number of factors that can contribute to flaring of your nose when smiling or animating your face. However, if you nose looks OK in repose (when not smiling), this is most likely do to the action of a very small muscle at the bottom of your nose. This muscle, the depressor septi nasi muscle, pulls the base of the nose back and to the sides when you smile.
As this is a small muscle this may be a little hard to hit with Botox. However, if other muscles of your face are contributing to this, the Botox may help. But, as always, this has to be done carefully so that muscles that you need will not be weakened too much. You need to be seeing an experienced injector for this.
This muscle can be divided surgically. Frequently I will do this with a rhinoplasty. It is fairly simple to do, but you will have some numbness of your upper lip for a couple of months.