Shirdi is accessible year-round, but the experience varies enormously by when you visit. The most practical advice is to avoid Thursdays, which is the most sacred day of the week for Sai Baba devotees — crowd numbers at the main temple can be three or four times higher than on other days.
Major festivals see extraordinary pilgrimages. Ram Navami in March or April and Guru Purnima in July typically draw hundreds of thousands of devotees over several days. Diwali also brings large crowds. If you want a quieter, more contemplative visit, come on a weekday in the dry season (November to February) and not on a festival day.
The climate in Shirdi is typical of the Maharashtra plateau — hot and dry from March to June, with the monsoon arriving in June and running through September. October and November are the most comfortable months weather-wise.