Is there a special reason you want this interpreted?
I am the only person in my household holding up the banner for Jesus Christ.
The passage you request is from the story of Paul and Silas in Philippi. Delivered from jail by an earthquake, they nonetheless remain in place so as not to cause their jailer to be blamed for their escape. The jailer is so impressed he immediately asks, "What must I do to be saved?" The reply is the verse you cite: "And they said, "Believe in the Lord Jesus, and you will be saved, you and your household."
We understand "saved" to describe the process by which we are released from our own sense of imprisonment in a cell of negative, limited thoughts and beliefs that create negative, limited life experiences, and a restoration to our innate freedom as the Christ—love-based, creative expressions of the Allness of God. This release from limited thinking is essential if we are to both create and experience the full Christ identity that Jesus demonstrates and calls us to. So it is by believing in that demonstration, and in the teachings on which it is based, that we are "saved." This is different from simply believing that Jesus is the Christ of God in expression. We must also come to believe that we ourselves are the Christ of God in potential, and that by following the teachings and example of Jesus, we are to become equally expressive of that Christ Power.
We also understand that each of us must come to that faith—we don’t "save" others. So this sentence could more clearly be stated as "Believe in the Lord Jesus, you and your household, and you will be saved." With regard to your statement that you are the only person in your household holding up the banner for Jesus Christ—that sounds dangerously like the kind of judgment that Jesus himself repeatedly urges us not to indulge in. We are never given the authority to pass judgment on anyone else’s spiritual journey. If we focus on our own, and truly express the infinite love that we will find in our own "salvation," we will be the light that Jesus calls us to be, and we will transform our lives, relationships and the world around us simply by being who we truly are, and by extending unconditional love to everyone around us.