The fans of the teams who make the NIT tournament certainly care about it. They will watch their teams try to make their way to the top. Everyone else will be too busy following the NCAA tournament to really give the NIT any attention. It's actually a good consolation for those teams that don't make the cut, and those that wouldn't stand a chance in the larger tournament.
Sometimes even good school teams do not make the NCAA tournament, and those teams certainly have their share of fans. The NIT tournament is their alternative to watch their team play in a tournament, even if it's not the main tournament. Sure, viewership won't be as high, but there will still be plenty of fans for it.
I do not think people care about NIT. I have never heard of the NIT Tournament. I like to think of myself as a socially aware person, and thus, if I cannot even begin to imagine what NIT stands for, I do not think many people are about the tournament.
I do not think people care about the NIT tournament because I am not sure that people even know what the NIT tournament is. I certainly don't. What is it? Is it sports? Is it poker? What is the NIT tournament, and why should I care about it at all?