You can ask that same question to anyone that cares what happens after they die. And although it might be a good question, nobody will stop caring.
You don't exactly survive by millions of people, because when you die you lose all your memories, experiences , thoughts and feelings. You won't be able to feel anything ever again, and it doesn't matter whether or not others might or might not remember you because you will not.
Technically, if you reproduce, your genetic memory will live on as long as your bloodline. I think it was Aristotle or Socrates that stated that every man seek immortality. A common man would go about this by having children an making sure his next generation could follow suit, but this was futile and at somepoint this man would be forgotten and lost in time. A great man would live on though his ideas an virtues. Also, how do you know what happens when you die?
If you have children they are not you, you may some of the same genes but you do not control then like you control your body, so they are not you. It still doesn't matter if your genetic memory will survive because you as a person will not. I do not know what happens after death but all the theories about after life or reincarnation are just ideas that just some people made, without any evidence. I think it is more likely to not feel anything like before you were born, rather than an idea that people made.
Children are exactly half of you. So they are definitely part you. An genetic memory is just as valid as your temporary ones except it last longer than you or your body.
I believe, if we were to collectively stop caring, we would be essentially be admitting defeat. Against what? Everything I suppose. Being human, some might say, is synonymous with caring... Otherwise, how have we made it to this point?