Why did the Dalai Lama seek asylum in India?

He sought asylum, as the situation in Tibet had become worse.

The Dalai Lama sought asylum in India in 1959 because the situation in Tibet had deteriorated due to the Chinese suppression of the uprising.
In 1958, there was an armed uprising in Tibet against China’s occupation. This was suppressed by the Chinese forces. Sensing that the situation had become worse, in 1959, the Dalai Lama crossed over to the Indian border and sought asylum which was granted.