What is the best time to visit Auli? Actually anytime, except during the monsoons.
There are two reasons for visiting Auli. One is for the magnificient view of the high peaks of the inner Himalayas. If the weather is clear you get an astounding panoramic view of famous peaks such as Nanda Devi, Dronagiri, Ghoda Parvat, Haathi Parvet, Mana Peak, Kamet etc. If you trek higher up from Auli to the Gorson Bugyal and right up to the highest point, the Gorson Top, the view is even better. So if the monsoon clouds are going to block your view of these peaks, it is understandable that the whole point of going to Auli gets lost.
In fact the best time for viewing and photography of the surrounding peaks from Auli is in September and October, immediately after the monsoons. By this time of the year the monsoon rains have cleaned up the atmosphere of haze and dust, thus giving us the ideal conditions for viewing of the distant peaks.
The other good reason for visiting Auli is for skiing. Auli and Gulmarg have the best skii slopes in India. In the case of Auli the best time for skiing is from mid-January till early March. You can just play around in the snow and get some good photographs, or if you are a serious enthusiast, you can opt for ski courses that can vary from 3 days to a fortnight. Ordinarily you can drive right up to Auli, but if the snow on the ground makes that difficult, the other option is to reach Auli by the cable-car, or Ropeway as the locals call it.
One word of caution. If you plan to drive to Auli using Google Maps for navigation, you are advised to read this previous post on the subject.