The standard answer to this one is Kanyakumari (also called Cape Comorin), which is indeed the southernmost point on the Indian mainland. Unlike the other extreme points, this is well populated and linked to other parts of the country. It is officially at 8.078 N, 77.541 E. Here is the main tourist area:
The southernmost point on the mainland would probably be the place marked “Hidden twin beach”.
The railway station seen on this map has trains to all corners of India, including Katra in Kashmir and Dibrugarh in Assam.
Kanyakumari was earlier in Travancore state, then Travancore-Cochin state and was finally transferred to Madras state (now Tamil Nadu) when linguistic states were formed.
But the true southernmost point of India lies in the Andaman and Nicobar islands (not the islands of Lakshwadeep as their southernmost point is around the same latitude as Thiruvananthapuram). To be precise, the southernmost point is a settlement called Indira Point on the southern tip of Great Nicobar island.
This article mentions that there are 4 households in this settlement with a total population of 27 (all males). It also mentions how many are literate and how many are from reserved categories. Presumably most (if not all) of them are connected with the lighthouse.
Here is a video of the area shortly after the tsunami of 2004: https://www.youtube.com/watch?v=XRA4sCNTN4E
Note how the lighthouse has sunk into the water. It is functioning again now.
There appears to be a helipad nearby. Also note the numerous small buildings near the lighthouse.
While tourism is being encouraged in some parts of the Andamans, even Indian citizens cannot visit Nicobar district without permission from the union territory’s government. There are ships and helicopter services to nearby Campbell Bay, which has India’s southernmost airport and naval base. This could be described as the southernmost village in India (population around 5700). Due to the military presence, it has amentities such as banks and a Kendriya Vidyalaya (school run by the federal government).
It is not often appreciated that these islands, particularly in the Nicobar group, are considerably closer to other countries than to India. This map should give an idea:
The straight-line distances from Indira Point to various places in India:
Port Blair-554 Km
And to some places in other countries:
Banda Aceh, Indonesia-215 Km
Phuket, Thailand-515 Km
Think about it. Indira Point is closer to Indonesia and Thailand than to Port Blair. And Yangon and Colombo are closer to it than any mainland Indian city.
The Andaman and Nicobar islands were occupied by Japanese forces during WW2, and were nominally governed by the Indian National Army.
Regarding the research stations in Antarctica starting with Dakshin Gangotri, it appears that India does not have any territorial claims over Antarctica while some other countries do.