This is actually an excellent question. And the Bible has no shortage of information to satisfy the answer. So, why should a believer be concerned with curbing sin since forgiveness and eternal salvation are secured through faith in Christ? I will give you seven reasons.
1. A life of sin is incompatible with the life of the Holy Spirit now indwelling us.
For the sinful nature desires what is contrary to the Spirit, and the Spirit what is contrary to the sinful nature. They are in conflict with each other... Galatians 5:17 (NIV)
A believer who gives in to a life of sin will be in constant conflict with the Holy Spirit who now indwells them. In a word, they will be miserable! (See Psalm 32:3-4 to see how David described this kind of conviction.)
2. Although it can be a difficult concept to grasp, the Bible says we have "died to sin."
The Apostle Paul wrote:
What shall we say, then? Shall we go on sinning so that grace may increase? By no means! We died to sin; how can we live in it any longer? Romans 6:1-2 (NIV)
When we come to Christ as Savior we literally enter into His death. That means we are now dead to the control of the old sin nature that once dominated our behavior. We are now free to follow Christ and make Him Lord of our lives. To live in sin would be a complete denial of the freedom Christ won for us on the cross. Living in sin means walking back into a life of slavery and bondage to the flesh.
3. Victory over sin is the sign that we have come to know Him and are born of Him
We know that we have come to know him if we obey his commands. 1 John 2:3 (NIV)
4. Living to please God enables us to test and approve God's will.
Do not conform any longer to the pattern of this world, but be transformed by the renewing of your mind. Then you will be able to test and approve what God's will is--his good, pleasing and perfect will. Romans 12:2 (NIV)
5. We owe Jesus our obedience, gratitude and worship.
Therefore, I urge you, brothers, in view of God's mercy, to offer your bodies as living sacrifices, holy and pleasing to God--this is your spiritual act of worship. Romans 12:1 (NIV)
6. Because sin always brings death
Don't be lulled into the idea that just because you're a Christian that sin can no longer produce death in your life. I have personally witnessed born-again believers who have suffered great loss because they became careless with allowing sin in their lives. Everything from the death of a marriage, family, and career can easily develop from a life of reckless ignorance.
For the wages of sin is death... Romans 6:23 (NIV)
...after desire has conceived, it gives birth to sin; and sin, when it is full-grown, gives birth to death. James 1:15 (NIV)
Put to death, therefore, whatever belongs to your earthly nature: sexual immorality, impurity, lust, evil desires and greed, which is idolatry. Because of these, the wrath of God is coming. Colossians 3:5-6 (NIV)
The acts of the sinful nature are obvious: sexual immorality, impurity and debauchery; idolatry and witchcraft; hatred, discord, jealousy, fits of rage, selfish ambition, dissensions, factions and envy; drunkenness, orgies, and the like. I warn you, as I did before, that those who live like this will not inherit the kingdom of God. Galatians 5:19-21 (NIV)
All in all, I think these biblical reasons for avoiding sin are compelling. Obviously, I could have come up with many more witnesses from God's Word, but I think the point has been sufficiently made. As a new creation in Christ, a life of sin no longer fits with our eternal destiny and blessing. In fact, it's the opposite of what God wants to do in our hearts.