In short, travel to India shouldn't be avoided just because its the monsoon season. For sure, it can make travel more difficult at times, and greater consideration needs to be given to packing and staying healthy during the monsoon.
However, it doesn't rain all day every day during the monsoon. Once the initial onset of the monsoon has passed, the rain usually settles down into a pattern of raining for at least a couple of hours most days. It's very unpredictable though, and sometimes there will be no rain at all for days. Also, some areas of India (such as the desert) receive less monsoon rain, and some (such as Leh and Ladakh) don't receive any monsoon rain at all.
Places such as Goa and the Valley of Flowers National Park are particularly green and beautiful during the monsoon, and are top destinations for monsoon travel in India. Many hotels also offer discounts on accommodations as well.