I want to know what guidance the Bible has for Christians who are witness to sin, and adultery specifically.
This answer points to the scripture only because that is what was requested.
Being a witness to adultery is really no different than being a witness to any sin. The Bible gives a blanket answer for all sins you witness a Christian brother or sister commit.
It is in Matthew 18, starting in verse 15, and it is commonly called the Matthew 18 Principle.
Matthew 18 (NIV)
15 “If your brother or sister sins, go and point out their fault, just between the two of you. If they listen to you, you have won them over. 16 But if they will not listen, take one or two others along, so that ‘every matter may be established by the testimony of two or three witnesses.’ 17 If they still refuse to listen, tell it to the church; and if they refuse to listen even to the church, treat them as you would a pagan or a tax collector.
Every Christian should desire not to sin. However, many of us do sin regularly and sometimes we continually commit the same sins. Matthew 18 covers this circumstance. So if the Christian brother in question committed this sin only once, the advice the Bible gives is not exactly clear what you should do, if anything. The key, though, in all circumstances is repentance. The sinner, compelled by his love for Christ, should desire to sin no more and turn from sin and evil ways.
Matt. 18 is a passage about forgiveness, however, is quite clear that you give a christian sinner a few chances to repent and if he does not then separate him from the body and call him a fellow believer no more.
Now we should note that the parable of the unmerciful servant immediately follows the Matthew 18 principle described above. Forgiveness for the repentant sinner is central to Christian doctrine.
I adapted this from a previous answer for a different question, but is not a duplicate