Somehow, someone got to his daughter and tried to turn her into an instrument of the Dark Lord so that she could be used to kill his pet Muggles.

Not to mention that Ginny's actions would reflect very badly on him. Between Arthur's reaction here, and Molly's vocal desire for her kids to have "respectable" jobs (she disapproves of the twins' joke shop), appearances (Bill's long hair and earring), etc., the Weasleys strike me like the kind of parents who measure their kids' achievements only by how well they reflect on their parents.