Scott asked a valuable question: Why should we care about these people?
He cited the passage on p. 154 regarding Tony's experience with the family history, which I agree is Mann's first explicit answer to the issue. We care because this family is convinced of its own standing and importance, and not in a self-important or ironic way. Scott's right that Tony's experience in this scene is touching and illuminating, for her and for us.
And this got me thinking about Buddenbrooks' structure. I believe Mann makes us care because, in the first third of the novel, at least, he positions the family's downfall as simultaneous with Tony's coming-of-age. As her relationship with Grünlich progresses through Parts 3 and 4, she goes from a strongwilled, rebellious teenager to a more resigned, dependent young mother. But when Grünlich's business fails and her father takes her back to the family home on Meng Strasse, she confesses that she's "learned something about life… One is no longer such a silly goose." (p. 234 in the Vintage edition.) This is small consolation for the family's embarrassment, which is no doubt the point. But Tony's undergone the most basic experience of any bildungsroman, and I think Mann draws us deeper into this troubled family's problems by layering his narratives in such a way.