I think the numbers are somewhat disputed. The People's Linguistic Survey of India says there are at least 780, with ~220 having died out in the last half century.[1] The Anthropological Survey of India reported 325 languages.[2]
The discrepancies are made particularly tricky because of the somewhat ambiguous distinction between languages and dialects.
I'm not an anthropological linguist, so I can't say with any authority. I only listed two figures there, but there are many more disputed figures.[1] There are a lot of difficulties classifying languages in India because the line between language and dialect can be tricky to pin down.