What is the biblical basis for the doctrine of Eternal Security? The Doctrine of Eternal Security (also called the Perseverance of the Saints) teaches that once a person truly experience salvation, that person's eternal destiny of eternal life is immutable.
What is the biblical basis or biblically-based argument for this doctrine?
Oh, how I dislike the expression &quot;once saved always saved&quot;!  It gives a wrong impression that if you just make some sort of “sinners prayer” and ask Jesus to accept you (like you’re doing Jesus a big favour), then you are saved.  Nothing else required.  Just pat yourself on the back and carry on.  Any such notion of “easy believism” is utterly false and misguided.
Let us be clear from the outset – salvation is a sovereign act of God whereby an unregenerate sinner is washed, renewed, and born again by the Holy Spirit:

At one time we too were foolish, disobedient, deceived and enslaved by all kinds of passions and pleasures. We lived in malice and envy, being hated and hating one another.  But when the kindness and love of God our Saviour appeared, he saved us, not because of righteous things we had done, but because of his mercy.  He saved us through the washing of rebirth and renewal by the Holy Spirit, whom he poured out on us generously through Jesus Christ our Saviour, so that, having been justified by his grace, we might become heirs having the hope of eternal life. (Titus 3:3-7).

Here are some Bible verses in support of the Christian assurance of salvation:

I will not leave you as orphans; I will come to you.  Before long, the world will not see me any more, but you will see me.  Because I live, you also will live.  On that day you will realise that I am in my Father, and you are in me, and I am in you.  Whoever has my commands and obeys them, he is the one who loves me.  He who loves me will be loved by my Father, and I too will love him and show myself to him. (John 14:18-21)


You, however, are controlled not by the sinful nature but by the Spirit, if the Spirit of God lives in you.  And if anyone does not have the Spirit of Christ, he does not belong to Christ.  But if Christ is in you, your body is dead because of sin, yet your spirit is alive because of righteousness.  And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you. (Romans 8:9-11)


And those He predestined, He also called; those He called, He also justified; those He justified, He also glorified. (Romans 8:30)


Who will bring any charge against those whom God has chosen? It is God who justifies. Who is he that condemns? Christ Jesus, who died—more than that, who was raised to life—is at the right hand of God and is also interceding for us. (Romans 8:33-34)


For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in all creation, will be able to separate us from the love of God that is in Christ Jesus our Lord. (Romans 8:38–39

There are plenty other Bible verses that confirm the assurance born-again Christians have of their eternal salvation.  But no person who turns round and denies Christ Jesus as their Lord and Saviour can say they were ever born-again by the new, spiritual rebirth or that they were indwelt by the Holy Spirit.
I hope the Scriptures I've quoted show that there's more to being a Christian than simply making some sort of &quot;sinners' prayer&quot; and saying you will accept Christ Jesus.  Commitment is called for.  Obedience is called for.  A transformed life is called for.  You have to BELONG to Christ Jesus and follow Him, regardless of the price.
Once the believer has experienced this rebirth the Spirit that indwells them gives assurance of their salvation.  They can cry out, &quot;Abba, Father&quot;, confident they have been adopted into God's family.  It's a done deal.
Edit: In answer to your comment, “can the done deal get undone along the way (theoretically)?” here is how it works.  Genuine repentance results in genuine forgiveness.  Once you get up off your knees and dry your eyes you are overwhelmed with the reality of God’s love.
Once God chooses the repentant believer (N.B. God does the choosing) he is justified, not by anything he does, but because God purposed it.  Is God about to change His mind and, further down the road say, “Woops, made a mistake with that one.  Better cross his name out of the Book of Life”?
The believer who has been reborn has been regenerated.  Is he to be un-regenerated because God got it wrong and didn’t know what was round the corner?  Where in the Bible does it say the new birth can be taken away?  Nowhere – so don’t bother looking.
John 3:15 says that whoever believes in Jesus Christ will &quot;have eternal life.&quot;  Not, “might have eternal life” or “if you slip up that’s it – you’re history”.  If you believe in Christ today and have eternal life, but lose it tomorrow, then it was never &quot;eternal&quot; at all. Therefore, if the born-again believer loses their salvation, the promises of eternal life in the Bible would be a pack of lies.
Jesus made this promise to all believers who turn to Him in faith:

And this is the will of him who sent me, that I shall lose none of all that he has given me, but raise them up at the last day.  For my Father’s will is that everyone who looks to the Son and believes in him shall have eternal life, and I will raise him up at the last day. (John6:39-40).

No if’s, not but’s, no hidden default clauses, no traditions of men to follow, no man-made rules to adhere to.  God keeps it simple.  Humans complicate things in order to control and manipulate.
I’m not about to call God a liar.  He is faithful and true and we can depend upon His promises.  Bottom line is that God is Sovereign, and He draws us to Him.  End of.
Concluding Summary:
I have been saved – in the past – from the penalty of sin – by a crucified Saviour: “For in this hope we were saved&quot; (Romans 8:24). “For it is by grace you have been saved, through faith – and this not from yourselves, it is the gift of God” (Romans 2:8).
I am being saved – in the present – from the power of sin – by a living Saviour: &quot;For the message of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God&quot; (1 Corinthians 1:18).
I shall be saved – in the future – from the presence of sin – by a coming Saviour: &quot;Since we have now been justified by his blood, how much more shall we be saved from God's wrath through him!&quot;  (Romans 5:9)