Who is responsible for our sanctification?
For those whom He foreknew, He also predestined to become conformed to the image of His Son, so that He would be the firstborn among many brethren;
— Romans 8:29
So then, my beloved, just as you have always obeyed, not as in my presence only, but now much more in my absence, work out your salvation with fear and trembling; for it is God who is at work in you, both to will and to work for His good pleasure.
— Philippians 2:12-13
But by the grace of God I am what I am, and His grace toward me did not prove vain; but I labored even more than all of them, yet not I, but the grace of God with me.
— 1 Corinthians 15:10
Now the God of peace, who brought up from the dead the great Shepherd of the sheep through the blood of the eternal covenant, even Jesus our Lord, equip you in every good thing to do His will, working in us that which is pleasing in His sight, through Jesus Christ, to whom be the glory forever and ever. Amen.
— Hebrews 13:20-21
Sanctification was always the goal of believers, both in the Old Covenant as well as the New. Grace is the way to sanctification, and grace was always given through faith, being loyal. However sanctification was not always possible.
The first instance of the possibility of God's people being sanctified was in the Exodus. However because of unbelief, refusal to consider the display of God's great works as proof that He would support them in the hard journey ahead, God swore that they would never enter His rest.
So how does rest equal sanctification?
Because no one is sanctified, separated, distinct from the rest of the world unless he or she is without blemish. And the only way that one can be without blemish is by being IN Messiah.
Luke 2
9And an angel of the Lord suddenly stood before them, and the glory of the Lord shone around them; and they were terribly frightened. 10But the angel said to them, “Do not be afraid; for behold, I bring you good news of great joy which will be for all the people; 11for today in the city of David there has been born for you a Savior, who is Christ the Lord. 12“This will be a sign for you: you will find a baby wrapped in cloths and lying in a manger.” 13And suddenly there appeared with the angel a multitude of the heavenly host praising God and saying,
14“Glory to God in the highest,
And on earth peace among men with whom He is pleased.”
So by withholding the sending of Messiah, Israel had been preventing from entering rest. And by sending Him when the remnant had pleased Him, it was now possible to abide in Him.
Again, abiding in Christ requires grace, which is sufficient, but to receive grace requires, not works of the law, but faith, loyalty. Paul showed loyalty by praying three times for his sinful weakness to be removed, as taught by God, but to prevent him from becoming proud, God allowed the weakness to remain. However it didn't prevent him from being sanctified, being in Christ, because God said His grace, given to the faithful, was sufficient, would do the job.
So we can see that faith, loyal response, from the believer, is required for sanctification.
As for predestination, it does not speak of God choosing individuals, but it speaks of the method: conforming to the image of Christ, by being IN Him. In other words, from the beginning of creation, God always planned to perfect creation, subdue the earth, by using humanity, by conforming them to Christ.