This one probably had more to do with protecting shipping lanes from French interdiction, given the location and which other countries had colonies and other operations semi-nearby (spoiler alert: France). As far as why the British cared to hold it in the last century, I mean, not for why it's still a base now. Leave all the islands in that area to France and it'd have been a dagger pointed at shipping crossing the Indian ocean via the canal. Seize a couple of the islands yourself, and at least you can keep an eye on things and mount a plausible threat if a hot war breaks out, to keep France from getting too bold with their use of the nearby islands.

Britain had a bunch of other islands much closer to India. Not even counting Sri Lanka.

Though, yes, protecting shipping lanes to and from India would have been part of the purpose, but not just to India. British Malaysia, various Pacific territories, and the commonwealth states of Australia and New Zealand, would have shared those same shipping routes, largely.

I'd guess that as a US base it's more likely to support US or NATO operations, real or hypothetical, in or around the Horn of Africa than it is to have much to do with the subcontinent.

For the US timeframe, the island was viewed during the Cold War as strategic to the United States, due to it's proximity to India, a potential ally of the Soviet Union. It served (and may still serve) as a Navy communication station. It has been used to monitor Afghanistan and China's activities in the South China Sea.