How do proponents of sola fide explain Titus 1:16? Evangelicals believe that faith alone establishes your salvation.  How do they resolve Titus 1:16?


  They profess that they know God; but in works they deny him, being abominable, and disobedient, and unto every good work reprobate.


While I understand that evangelicals don't directly recommend doing abominable, disobedient and reprobating good work - they also generally don't believe that you can lose your salvation, which this passage seems to imply.
Laying the Groundwork

The Transformation of Salvation

It is important to note that, according to the Bible, the following things occur at the moment of salvation by faith alone:


We, who were once dead, are made alive in Christ:


  When you were dead in your transgressions and the uncircumcision of your flesh, He made you alive together with Him, having forgiven us all our transgressions.  (Colossians 2:13)

We, who were once in the dominion of darkness, are brought into the kingdom of Jesus


  For He rescued us from the domain of darkness, and transferred us to the kingdom of His beloved Son, 14 in whom we have redemption, the forgiveness of sins.  (Colossians 1:13-14)

We, who were once alienated from God, hostile to God (or enemies of God) have now become the children of God by virtue of a new birth:  


  And although you were formerly alienated and hostile in mind, engaged in evil deeds" (Colossians 1:21)
  
  But as many as received Him, to them He gave the right to become children of God, even to those who believe in His name, who were born, not of blood nor of the will of the flesh nor of the will of man, but of God.  (John 1:12-13)

We, who were one sinful, are now a holy nation, made righteous and renewed in our spirits:


  But you are a chosen race, a royal priesthood, a holy nation, a people for God’s own possession, so that you may proclaim the excellencies of Him who has called you out of darkness into His marvelous light.  (1 Peter 2:9)
  
  For all have sinned and fall short of the glory of God, 24 being justified [declared righteous] as a gift by His grace through the redemption which is in Christ Jesus.  (Romans 3:23-24)
  
  Do not lie to one another, since you laid aside the old self with its evil practices, 10 and have put on the new self who is being renewed to a true knowledge according to the image of the One who created him.  (Colossians 3:9-10)



The Impact of the Transformation

So, at the moment of salvation, a very significant transformation has occurred.  We are changed at the very core of our beings so that our desires have changed and our spirits are now alive.  Indeed, we have the Holy Spirit Himself indwelling us:


  Or do you not know that your body is a temple of the Holy Spirit who is in you, whom you have from God, and that you are not your own? (1 Corinthians 6:19)


What this results in is a drastic change in desires.  Someone who has truly put their faith in Christ should have new desires, and their salvation should naturally result in a change in their actions.  

So, you should see the impact of their faith in the works that they do, but it is not a salvation by works.  Put another way, we believe that salvation is by faith alone, but a genuine faith will inevitably result in a change in actions and behavior.  If you observe no change in desires or actions, then it is cause to doubt the genuineness of the faith.

Titus 1:16

From an evangelical perspective, the verse you cited is not speaking about those who truly believe in Christ, but whose actions have failed to change.  Rather, it is speaking of those whose faith is not genuine and whose actions reveal this to be true.  Jesus Himself indicated that "by their fruits you will know them" in Matthew 7:16.

Conclusion

So, if you observe someone professing faith in Christ who is doing abominable things without shame or repentance, we should not conclude that this is someone who has truly put his or her faith in Christ, been brought out of the kingdom of darkness into the kingdom of light, made a new creation, forgiven, cleansed, made righteous, made alive and indwelt by the Holy Spirit of God, but who is failing to reflect in any way the life of a Christian.  Rather we should conclude that no change has really taken place and, as a result, no true faith has been exercised.