I'm a white British woman raising my kids (biracial) in an Asian country---though not South Asian. So I guess a little bit the other way round, in some ways!
I would always urge people to do whatever they can to teach their children their own minority language and culture to their children--even if they don't end up "perfect" speakers (or any kind of speaker, actually), even if they seem uninterested some of the kind.
Trevor Phillips - the guy who has been a significant and interesting voice on issues of minority culture and identity these last few decades - has talked about something called "cultural protection" which I find very interesting. His opinion is the minority groups do best within their societies if they can manage to have the right level of cultural protection, not too much or too little. Too much cultural protection is where a minority culture become ghettoized and is unable to succeed on the terms of their wider society. But too little cultural protection is where a minority group is too eager to throw away everything that makes them special and distinctive in comparison to the mainstream culture, and tries to assimilate completely, often believing idealistically that "If we assimilate hard enough, we'll be treated just the same, right?" This can result in what is sometime called "downward assimilation", where the minority group in question assimilates into the least-educated and most economically disadvantaged level of the mainstream culture, taking on problematic aspects of this group.
I can see what he means, and it makes sense to me. Anyone who is a member of a racial minority is always at risk of racism and prejudice, the whole thing of "I have to do better than the people around me in order to be considered not-quite-as-good." Having some special skills, an extra language, something that you can feel proud of as distinctive and different, some ethnic networks to help you out at times, can help you to "make up" for these problems, and be helpful to you in navigating a society that can sometimes be a more hostile place for you than it is for members of the mainstream ethnic group.
Here in Japan, I've known a few cases where an English speaking non-Asian (white or black) parent has done nothing to teach their language or culture to their children. Their children still get the "othering" treatment, annoying questions and patronizing assumptions that come with being a visible ethnic minority person, but they have no special strings to their bow that they might feel proud of to help them get through the difficult moments. They just wind up being a "Japanese kid with brown hair (or brown skin and curly hair) whose face doesn't fit." It's a bit of a crap situation.
The other thing is that it's OK if kids don't speak their minority language perfectly, or even speak it at all, or if they don't show any interest in their minority culture when they are young. As long as they have a base, they can pick up with the language and improve once they get older and develop an interest. So many kids develop an interest in their root in adolescence, or when they go to uni, or if they decide to go traveling. Even being a passive bilingual (who can understand a language but has never really spoken it) is an excellent platform for getting properly fluent with the language later on. I've known quite a few people who say things like "I wish my parents had spoken XZY to me." I've never actually met anyone who says "My parents did speak XYZ to me, and I wish they HADN'T, what a waste of time that was."
Also, looking at the state of the UK right now, it occurs to me that it' great to give kids a potential base in another culture. Right now, the forums are full of people saying "I'd tell my kids to emigrate, but they don't speak any languages other than English...." There is a lesson in there, I think!
Phew, I seem to have written a novel!