What sort of influence should the Bible have on society?

TL;DR:

The Bible influences society as believers live according to its precepts. Following God and His Word leads to the best outcome for any society.

Clearly God's Word is powerful. As such, it makes sense that it should influence our societies. The question is what is exactly meant by "influence" and by "society"? Israel in the Old Testament was a theocracy, but God nowhere tells Christians to try to establish a similar form of government. In fact, Christians are told to submit to the governments under which they find themselves (Romans 13:1–7). At the same time, this does not mean that we should not attempt to influence our governments to rule more in line with the truth of God's Word. Also, influence in a society is about far more than governmental structure. The Bible should influence the way Christians live, as such it should be influencing the societies in which we live. The more people in our societies know Christ and live godly lives, the more influence the Bible will have on societies.

from the old testament

Isaiah 55:10–11 talks about the effectiveness of God's Word. It will accomplish the purposes God has for it.
Psalm 33:12–22 says this: "Blessed is the nation whose God is the LORD, the people whom he has chosen as his heritage! The LORD looks down from heaven; he sees all the children of man; from where he sits enthroned he looks out on all the inhabitants of the earth, he who fashions the hearts of them all and observes all their deeds. The king is not saved by his great army; a warrior is not delivered by his great strength. The war horse is a false hope for salvation, and by its great might it cannot rescue. Behold, the eye of the LORD is on those who fear him, on those who hope in his steadfast love, that he may deliver their soul from death and keep them alive in famine. Our soul waits for the LORD; he is our help and our shield. For our heart is glad in him, because we trust in his holy name. Let your steadfast love, O LORD, be upon us, even as we hope in you." Following God leads to the best outcome for any society.

from the new testament

Even in the first century we see the impact the gospel had on a society. Acts 17 talks about Paul and Silas in Thessalonica. Some of the Jews, many of the devout Greeks, and some of the leading women of the city believed the gospel. Others were jealous, formed a mob, and attacked the home of Jason, where Paul and Silas were staying, to try to bring Paul and Silas out. But when they couldn't find Paul and Silas, the jealous men instead "dragged Jason and some of the brothers before the city authorities, shouting, 'These men who have turned the world upside down have come here also'" (Acts 17:6).
Christians are salt and light in the world (Matthew 5:13–16).
The Bible is powerful. It helps us to see rightly who we are before God. Hebrews 4:12–13 says, "For the word of God is living and active, sharper than any two-edged sword, piercing to the division of soul and of spirit, of joints and of marrow, and discerning the thoughts and intentions of the heart. And no creature is hidden from his sight, but all are naked and exposed to the eyes of him to whom we must give account."
Christians are told to submit to the governments under which they find themselves (Romans 13:1–7).
Believers know the Bible to be the true Word of God and seek to align our lives to what it says. We also have the indwelling Holy Spirit who is transforming us to be more like God (2 Corinthians 3:18). As we grow in our walk with Christ, we bear fruit (Galatians 5:16–26). This fruit has an impact on those around us. As the Bible influences us, it influences society.
Christians are also called to share the truth of God's Word with others (Luke 10:2; Matthew 28:18–20; 1 Peter 3:15–17). People need to know Jesus as their Savior more than they need to live in societies ruled by God's moral laws.

implications for today

The Bible should be the foundation for how believers live and interact in society. Since it is God’s Word, it should influence our values, actions, and attitudes. When we let Scripture shape us, we bring God’s love, justice, and truth into our society by the way we work, treat others, and make decisions. Living out biblical principles—like loving our neighbors, serving those in need, speaking the truth in love, and seeking peace—allows us to live as lights and salt in the world. It invites others to see God’s goodness. While societies may not ever become theocratic, using God’s Word as their moral and leadership guide, the way we live as pilgrims and sojourners in this world matters. We must learn how to live in the world but not of it, influencing our areas of influence around us with the Bible. Doing this allows us to bring light in a society that is otherwise driven by worldly desires and thinking. As believers live according to God’s Word, we can transform our environments and contribute godly values that reflect His grace and righteousness to society.

understand

Following God’s ways through God’s Word leads to the best outcome for any society.
Society is ruled by worldly governments, but Christians can impact society by living out kingdom values.
We must learn how to live in the world and not of it.

reflect

How does your life reflect the Bible's influence, and how does that impact those around you?
How are you allowing God's Word to shape your values, even when society promotes different ideals?
How can you let Scripture influence your actions and decisions in areas like work, family, and other societal involvement?

engage

The second president of the United States, John Adams, wrote, "Suppose a nation in some distant Region should take the Bible for their only law Book, and every member should regulate his conduct by the precepts there exhibited! Every member would be obliged in conscience, to temperance, frugality, and industry; to justice, kindness, and charity towards his fellow men; and to piety, love, and reverence toward Almighty God . . . What a Eutopia, what a Paradise would this region be" (Diary and Autobiography of John Adams, Vol. III, p. 9). Indeed, if we all attempted to live by the holy standards to which the Bible called us, our societies would be in much better condition.
For those Christians who live in nations where they can contribute to the process of selecting leaders and determining laws, the Bible should influence how they vote. For example, we know from the Bible that all of humanity is created in God's image (Genesis 1:27), thus it makes sense for Christians to seek legislations that value human life. In all the ways Christians are given opportunity to influence society, that influence should be directed by the truths of the Bible. How would society change if more people genuinely lived out biblical principles?
How can we support each other in standing for biblical values, especially when society may resist them?