At the beginning of Captain Phillips we see some scenes of him together his wife Andrea, driving to the airport and talking about their children. Later his family is mentioned rather rarely, he sends an e-mail to Andrea and at the end writes a letter to his family and speaks to them before his supposed execution.
But as Wikipedia says, the book on which the movie is based (and which is in turn based on real events) "alternates between Phillips' five-day ordeal and the plight of his family in Vermont, watching the drama unfold on cable news". While I, not having read the book, don't know how important or prominent this sub-plot about his family is in the book, the movie doesn't contain any treatment of what is going on in the outside world and especially his family, apart from the Navy ships.
The only reason I could imagine for the viepoint of his family to be dropped from the story might be to not interrupt the intensity of the hostage crisis (while showing his family could in turn infuse other emotions). But is there any official word why this sub-plot has been completely removed from the movie?