There's no correlation: Singapore and HK have some of the highest population density in the world, and have a low crime rate. Checking the list of countries by density[1] and then by violent crimes[2] will confirm that. India is quite violent, but for different reasons.
[1] https://en.wikipedia.org/wiki/List_of_countries_by_populatio...
[2] https://en.wikipedia.org/wiki/List_of_countries_by_intention...