「～さえ～ば」 is used to express a sufficient condition: "as long as ...", and 「体裁を保つ」 means "to keep up appearances", "to maintain a good public image", "save face", etc.
The line 「父が体裁さえ保てていれば家庭に無関心なこと」 more or less translates to:
(that) as long as he can keep up his appearances, my dad doesn't care anything about family matters
The translation might be potentially a little bit ambiguous as to its meaning, but the import of the line is not that the dad's neglect of his family enables him to save his face (that would make no apparent sense, as you observed), but that he is a usually absentee father (perhaps with a high-profile job) who would decide to get involved in family matters only if not doing so would jeopardize his precious, all-important public image. (Like paying a store manager a fair sum so that word doesn't get out that his son shoplifted a magazine or something. Or more subtle things like playing a good, hands-on father while people are watching -- it's a pretty common trope in fiction, though I'm sure there are a lot of instances in real-life too.)