Two things to consider in answering this question are: 1) applicable local, state, and federal laws in light of what Scripture says and 2) the business owner's conscience in light of what Scripture says.
Should Christian businesses provide services for gay weddings? When anti-discrimination laws require a Christian business owner to contribute to a gay wedding, what should the owner do?
The overarching principle to be followed in any walk of life is that God's Word takes preeminence over our own views and consciences as well as laws established by governments. It is evident that homosexual behavior is sin. Scripture is also more than clear that Christians must obey the laws of human authorities appointed over them. However, there are instances when God's children have disobeyed when those laws run contrary to His will. This article on civil disobedience gives examples in Scripture when going against a government's laws was appropriate, and provides guidelines to consider when contemplating doing the same.
Concerning our consciences, the Bible does not overtly forbid gay marriage. But it is clear that homosexuality is tied to a sinful nature (Romans 1:26-28); Scripture indicates that homosexual behavior is a symptom and natural outcome of an already depraved heart that has turned away from God. An argument can be made that contributing in any way to an event that strays this far from what God intended for mankind is foolish at best and sinful at worst (Romans 1:32). Others may contend that it could provide an opportunity to display God's love to those who are perishing. After all, Jesus ate with those upon whom religious leaders looked as being in a special category of sin, or the worst of the sinners. This article explains why Jesus met with those who seemed to need Him most. We must remember, though, that there is no indication that Jesus participated in actual ceremonies or rituals that were considered an abomination to God. He did not throw in His lot with them to sin. So, while He befriended these people, arguably the outcasts of society, His conduct was always above reproach. We are called to do the same thing. The Apostle Paul clarified his stand on how we should interact with unbelievers in 1 Corinthians 5:9-13:
I wrote to you in my letter not to associate with sexually immoral people—not at all meaning the sexually immoral of this world, or the greedy and swindlers, or idolaters, since then you would need to go out of the world. But now I am writing to you not to associate with anyone who bears the name of brother if he is guilty of sexual immorality or greed, or is an idolater, reviler, drunkard, or swindler—not even to eat with such a one. For what have I to do with judging outsiders? Is it not those inside the church whom you are to judge? God judges those outside. 'Purge the evil person from among you.'
With this balanced approach, Christians also need to realize that our actions must never even hint at sin. Ephesians 5:3 states, "But sexual immorality and all impurity or covetousness must not even be named among you, as is proper among saints." For many Christians, catering to a gay wedding would be crossing the line of indiscretion and opening them up to a bad name among other Christians—if not as ones who practice sin, as ones who condone the sin of others. This is where matters of conscience come into play. These same Christians also would understandably object to promoting a world view that Scripture says in reprehensible to God.
Christians who agree to supply services to a gay wedding have a number of options. They can quietly go about their business with excellence, get paid, and move on. They can also decide to try to turn the episode into one that advances the Kingdom of God. This approach would bring with it some exceptional challenges. First, knowing that God's Word never returns to Him void (Isaiah 55:11), to explicitly present the Gospel of Christ at such an event would still be unlikely to wield much of anything except an assumption of condemnation. Likewise, to express any reservation or warning about what the gay couple is doing would almost certainly be viewed as an antagonistic reproach. However, God may be at work in either of these. On the other hand, to lavish hospitality with one's services and joyously wish the couple well could easily be taken as a sign of accord with them and the ceremony. So, there is a fine line on which to balance when Christians choose to 'comply with' anti-discrimination laws and cater to homosexual weddings and other pro-gay events.
On the other side of the coin, once Christians decide prayerfully that civil disobedience is the road to take, they simply must be prepared to endure the punishment of the State. In the case of a small business such as a photographer, a local baker or florist refusing to cater to a gay wedding, the fines imposed are ostensibly meant to drive them out of business, if infractions continue unabated.
Considering all options, Christians should have the primary goal of honoring and glorifying God, who gives everyone—believers and non-believers—life. We also need to show love to everyone, and allow God's love to govern our every decision.
Where should we, as Christians, stand on gay marriage?
What does the Bible say about civil disobedience?
Is homosexuality a sin?
How can I seek first the kingdom of God?
What is the key to knowing the will of God?