Evacuations were ordered in low-lying areas off the coast of Tamil Nadu state, said Jatindranath Swain, the province's disaster management commissioner.
A ship with 37 people on board became grounded as the storm made landfall, he said. No immediate details were available about the type and origin of the stuck vessel, but Swain said naval divers were engaged in the rescue.
According to the latest bulletin issued by the India Meteorological Department (IMD), Nilam crossed the southern coast at 4:30 p.m.
Governments in neighboring Andhra Pradesh state and the federally-administered territory of Puducherry said they were prepared to deal with Nilam, but no evacuations had been ordered in those areas so far.
For now, disaster management officials have ruled out the possibility of a major loss of life from the storm that the country's weather office warned was packed with high-speed winds.
However, the national weather department has forecast heavy rains in parts of Tamil Nadu, Andhra Pradesh and Puducherry.
"Gale wind-speed reaching 80-90 kph (50-56 mph) gusting to 100 kph (62 mph) would prevail along and off north Tamil Nadu, Puducherry and adjoining south Andhra Pradesh coasts during next 24 hours," it said.
Weather officials also predicted a storm surge that was likely to inundate low-lying areas, mainly along Tamil Nadu's coast.
Forecasters warned that residents should expect extensive damage to thatched roofs and huts from Nilam and advised fishermen not to venture into the sea until the storm has passed.