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.
