Should biblical laws apply to non-Christians? Obviously a lot of Christians want to have biblically based laws in the society, in effect requiring everybody to follow biblical morals. But the opposite is also a popular view among Christians: as non-believers are not saved either way, we shouldn't make it our problem whether they sin little or much.

Which is it, and why?
It's unclear to me whether you're asking for an answer that relates to political or governmental policy, or how we ought to behave as individual Christians.  But to address the latter, 1 Corinthians 5:12-13 I believe is relevant:


  12 What business is it of mine to judge those outside the church? Are you not to judge those inside? 13 God will judge those outside. “Expel the wicked person from among you.”


In other words, regardless of whether Biblical laws would be helpful for non-Christians, as Christians it's not our place to pass judgment; we ought to leave that for God.