As an American, I personally don't care about the family. I don't actively follow them, but in a passive sense I do maintain some awareness of what's going on with them. I respect the good things they do with their status, as well as what service they seem to provide to the UK.
Individually, Americans may not care a whole lot about the royals or think that they're necessary. Looking at our own political history and pop culture, I think our tendencies paint a different picture. Politically, we seem to place a lot of importance on families and dynasties, most significantly with the Kennedy family. Hell, JFK's administration was referred to as Camelot. Even with the Bush family, much has been discussed about that family's role and influence. We constantly look to the young members of those types of families with the expectations that they will become leaders. We do the same with celebrities, placing power couples and families up on pedestals. We give these bloodlines in politics and entertainment so much more importance than what is actually there. So, regardless of what any American may say to the contrary, we as a people care, not so much about the royal family itself, but the notion of a royal family. It's something we've never really had, and I think it's the one thing we've missed from our British cousins.