The Mongols did invade India, even if they failed to progress beyond northern India. They fought against the Delhi Sultanate and even managed to sack Lahore in 1241.
The climate explanation is plausible at first, but doesn't make sense once you examine it further. The Mongols invaded southern China, Vietnam, and parts of Southeast Asia. These areas are all known for the humidity and heat. The more likely explanation is that, once Mongke died, Mongol territorial expansion grounded to a halt from internal infighting. Kublai, Berke, Hulagu, and Ariq Boke were more focused on fighting each other than continuing their expansion into India or anywhere else. Once the civil war settled down, the Mongol empire became too fractured to ever mount united campaigns like they were able to under Mongke or Ogedei.
This question is given in the FAQ section you can check it out Here
Actually, according to this source this read like cracked, so I am not very confident of it's veracity - The mongols could have come to India to invade as Gengis Khan was chasing the son of a king he defeated, who fled to india. The son was denied shelter by the reining king at the time called Iltutmish, so Khan didn't come after him in India. If iltutmish had granted hi shelter and allied with him, gengis khan would have surely come down in india and indian civilization would surely have been very different now.
Also take a look at
Sorry for not providing more strong sources. If anyone else can, that would be great :)