What is the biblical basis for the idea that people can't stop sinning?

In Matthew 19 it says,

16 Just then a man came up to Jesus and asked, “Teacher, what good thing must I do to get eternal life?”

17 “Why do you ask me about what is good?” Jesus replied. “There is only One who is good. If you want to enter life, keep the commandments.”

18 “Which ones?” he inquired.

Jesus replied, “ ‘You shall not murder, you shall not commit adultery, you shall not steal, you shall not give false testimony, 19 honor your father and mother,’c and ‘love your neighbor as yourself.’ ”

20 “All these I have kept,” the young man said. “What do I still lack?”

21 Jesus answered, “If you want to be perfect, go, sell your possessions and give to the poor, and you will have treasure in heaven. Then come, follow me.”

22 When the young man heard this, he went away sad, because he had great wealth.

23 Then Jesus said to his disciples, “Truly I tell you, it is hard for someone who is rich to enter the kingdom of heaven. 24 Again I tell you, it is easier for a camel to go through the eye of a needle than for someone who is rich to enter the kingdom of God.”

25 When the disciples heard this, they were greatly astonished and asked, “Who then can be saved?”

26 Jesus looked at them and said, “With man this is impossible, but with God all things are possible.”

The key phrase is "if you want to be perfect". The Greek word hamartia is the one used by the New Testament writers to indicate sin. It does not just mean doing what is wrong, but means "to miss the mark", like in archery. Any failure to do good is to miss the mark. So since the rich young ruler did not give away all his goods to the poor, he was missing the mark; he was sinning, he was not perfect.

Jesus renders Heaven's verdict about mankind: "With man this is impossible."

1john 2:1 says I write unto you that you sin not! 1 John 3:3 says we can be as pure as Jesus! 1 John 3:6 says anyone who abides in Christ sins NOT! 1 John 3:8 says he who sins is of the devil! The bottom line is sin is not characteristic of a true believer. Those who cannot stop sinning but rather justify and defend sin are not right with God!

Paul says in Romans 7:15, "I do not understand what I do. For what I want to do I do not do, but what I hate I do." What Paul "hates" to do here is referring to the sins against God that he commits. Also, take note that Paul was quite a godly Christian; if he can't stop sinning, then we probably can't either.

Another discussion point to have with O.P.'s "someone" are the following words of Jesus to Peter: "The spirit is willing but the flesh [body] is weak" (Matthew 6:41).

The apostle Paul is probably the best place to start in answering this question. In the student Bible I have, the passage title for Romans 7:7-25 is called "struggling with sin".

The verses you possibly are seeking are verses 14-25 (using NIV translation).

14. We know that the law is spiritual; but I am unspiritual, sold as a slave to sin.
15. I do not understand what I do. For what I want to do I do not do, but what I hate I do.
16. And if I do what I do not want to do, I agree that the law is good.
17. As it is, it is no longer I myself who do it, but it is sin living in me.
18. I know that nothing good lives in me, that is, in my sinful nature. For I have the desire to do what is good, but I cannot carry it out.
19. For what I do is not the good I want to do; no, the evil I do not want to do--this I keep on doing.
20. Now if I do what I do not want to do, it is no longer I who do it, but it is sin living in me that does it.
21. So I find this law at work: When I want to do good, evil is right there with me.
22. For in my inner being I delight in God's law;
23. but I see another law at work in the members of my body, waging war against the law of my mind and making me a prisoner of the law of sin at work within my members.
24. What a wretched man I am! Who will rescue me from this body of death?
25. Thanks be to God--through Jesus Christ our Lord! So then, I myself in my mind am a slave to God's law, but in the sinful nature a slave to the law of sin.

The doctrine you are addressing is called Total Depravity.

This question will actually say much more about it: What is the Biblical basis for Total Depravity?

Various denominations have slightly different nuances to the idea, but typically the following verses are first up:

Romans 3:10-11: "There is none is righteous, no, not one; no one understands; no one seeks for God."

Romans 8:7-9: "For the mind that is set on the flesh is hostile to God, for it does not submit to God's law; indeed, it cannot. Those who are in the flesh cannot please God. You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him."

Ephesians 2:3b: "[We] were by nature children of wrath, like the rest of mankind."

1 Corinthians 2:14: "The natural person does not accept the things of the Spirit of God, for they are folly to him, and he is not able to understand them because they are spiritually discerned."

In the Old Testament, one looks to:

Jeremiah 17:9 (NLT) "The human heart is the most deceitful of all things, and desperately wicked. Who really knows how bad it is?

Jeremiah 13:23 (NIV): "Can the Ethiopian change his skin or the leopard its spots? Neither can you do good who are accustomed to doing evil."

Basically, the idea is related to original sin, and says that mankind is incapable of not sinning.