I Corinthians 5:9-11 tells us not to associate with the sexually immoral who call themselves "brothers," though it does not say the same for those who do not claim to be brothers (and sisters, I presume) in Christ.
Leviticus 18:29 tells the people of Israel, regarding all sexual sins, to ostracize and cut all ties with any such people, whether they be native Israelites or visiting foreigners.
Leviticus 20:13 tells the Israelites to execute homosexuals.
These are three very different approaches to the same problem, and I think that the differences have everything to do with context. The difference between Leviticus 18 and 20 seem to be that one is a command to the citizens on their own proper response, and the other is a penal code for the governing authorities. Neither one likely applies to our situation, exactly, though I highly doubt that God's sentiments on the matter have changed any. There was a certain advantage to having a religious uniformity to the culture, which made Lev. 18 possible, and having a government-established religion made Lev 20 possible. The passage from I Corinthians adds the explanation for why we don't refuse to associate with such people who don't claim to be believers, "since then you would need to go out of the world." Simply stated, it isn't practical. Besides that, the kingdom of Heaven is something of a parallel to the kingdom of ancient Israel: both are a theocracy, and the laws apply to its members and those within its territory. Because our government is only God, himself, we leave the penalty of the sin to him, but we do our duty to not associate with sexually immoral people who would come within our territory, so to speak, and claim to be one of us.
I take the Joshua 24:15 approach. As for me and my house, we will serve the Lord. I will not call two men "married," and I will not call them "husbands." When my wife's cousin "marries" another man, that does not make the other man a member of my family. He is not a relative. He does not get invited to family events. He does not get Christmas cards. I don't hate him. I don't even dislike him. I just refuse to do the slightest thing to acknowledge any validity to the relationship. At work, I get along with such people as I would any other coworker. If I'm not pressed on my views, then I keep the conversation politely work-related. If pressed on my views, then I give the honest answer that they deserve, even though it will probably get me into trouble some day. I believe that they're all going to Hell unless they repent and turn from their sin. That's all there is to it.