I have a friend whose graduate program director really did have connections at his top choice and undoubtedly played a role in his acceptance there. He's going MD/PhD though and connections are more important given the smaller applicant pools.
As far as what you're asking, these connections are unlikely to somehow bring in the grant money. The thing about connections is that they only work if you know about them; you're a small fish in a big pond and nobody's going to be secretly pulling strings for you. If you're not the networking type I wouldn't worry that much about it anyway, because there's no guarantee that you will find some powerful benefactor.
Just get the pubs and do what excites you.