If we believed that Christianity was merely a good way to live, then there might be a justification for simply adhering to a "live and let live" philosophy such as it pertains to society and social issues. But it's not; rather Christianity is the expression of God's plan of salvation for a fallen creation, and our moral code is the expression of God's perfect and absolute moral law.
As Christians, we are called to be the "salt of the earth" and the "light of the world":
13 “You are the salt of the earth. But if the salt loses its saltiness, how can it be made salty again? It is no longer good for anything, except to be thrown out and trampled underfoot.
14 “You are the light of the world. A town built on a hill cannot be hidden. 15 Neither do people light a lamp and put it under a bowl. Instead they put it on its stand, and it gives light to everyone in the house. 16 In the same way, let your light shine before others, that they may see your good deeds and glorify your Father in heaven.
Salt is a natural preservative, slowing down the spoiling of food - it's not about changing the taste of the food it's about inhibiting the speed at which it rots. Additionally, a light dispels darkness - where there is light, darkness ceases to reign and dominate.
The effect of Christians on society as a whole should be to improve society as a whole. This mandate is, therefore, a command to have an impact on society, drawing general society to be a better reflection of God and his law, not simply a call to evangelize. Part of that is to have a social and political impact on society.
Furthermore, there is ample evidence in scripture that entirely reprobate societies are subject to God's judgement (witness Sodom and Gomorrah, the Noahdic flood, the Canaanites, etc). It is also worthwhile to examine the rise and fall of all empires in history; nearly all mark the end of their influence with reprobate societies, in particular of a sexual nature.
Apart from that, Christians have historically been deeply involved in society and social issues. This is because Christians "impose" their "values" onto those around them - specifically that all humans have inherent dignity and inherent "God-given" rights in keeping with that dignity.
So, I would say that Christians have a moral duty and obligation to have an impact on their society. Being involved in social-political, including law-making, issues is a part of that.
Now that being said, we do have to be careful of exactly what stand to take and how we go about taking it.