Do we need to constantly be asking for the forgiveness of sins after we've been born again in Christ?

Asking for forgiveness...

There is nothing wrong in asking for forgiveness if you know you have fallen into sin or had a moment of weakness. It is completely natural to want to apologize for your wrongdoings, but, realize that this does NOT ever affect your positional stance with God. Through a faith response to the grace gospel of Jesus Christ, and His cross work, you are once and forever cleansed from sin. As grace believers, our sins have been nailed to the cross of Christ, and we have, by the grace of God, received in its place, the imputed righteousness of Christ. Nothing can change our position in Christ and before the Father, so asking forgiveness, or not asking forgiveness, does not change or affect our standing before God.

The grace believer's salvation is forever secure and they are "sealed with the Spirit." (Rom. 5:1-2, 8:29-39, 6-8; Eph. 1:13-14, 4:30; Phili. 1:6; 2Tim. 1:12).

"In whom ye also trusted, after that ye heard the word of truth, the gospel of your salvation: in whom also after that ye believed, Ye Were Sealed With That Holy Spirit Of Promise," (Ephesians 1:13)

Upon our salvation, we are "sealed with the Holy Spirit" unto the Rapture (cf. Eph. 1:13, 4:30) or in death. God does not want you as a believer in Christ to doubt your salvation. Our salvation by grace does not depend on our performance or even our faithfulness. The salvation we enjoy today as believers is solely dependent upon what Christ has already done for us at Calvary's cross!

The act of asking forgiveness is a thing that we as humans need to do. It is a mental release; it is a way of showing sorrow and remorse. When we fail each other, we make up by asking forgiveness. So, in this sense, we do this in our relationship with God too, but even while we practice this behaviour, we should know that it does not change who we are positionally before Him. Since the death and resurrection of Christ, God sees us as perfect, as new creatures, spotless and blameless in His sight. (See my post on sanctification for more clarity on this topic.)

About being "born again"...

In addition to the above, I’d like to share another truth with you. It might seem small and insignificant, but nevertheless, it is more accurate when it comes to understanding our relationship with God in this grace dispensation.

We are not “born again” in Christ. The gospel of grace regenerates our spirit and we are made alive in Christ. Those that are “born again” are Israel. They were born once when they passed through the Red Sea at their exodus. After rejecting their Messiah, they will eventually be “born again” when they receive the outpouring of the Spirit of God upon them just before they enter into the Millennial Kingdom. Israel is ‘born again’. The member of the Body of Christ is regenerated in Spirit to become a new creature. It is important to remain accurate in our understanding and speech, to gain a complete image of who we are in Christ.