What's the biblical basis for non-believer's ability to do good? This is a bit of a theoretical question, but what is the biblical basis for the belief that non-believers can do good (doing good meaning performing a good deed from a good intent)?
On the contrary, scripture seems to show that humanity is incapable of doing anything good:

Genesis 6:5 ESV, Matthew 23:37 ESV The LORD saw that the wickedness of man was great in the earth, and that every intention of the thoughts of his heart was only evil continually. ... For as were the days of Noah, so will be the coming of the Son of Man.
Romans 3:11-12 ESV None is righteous, no, not one; no one understands; no one seeks for God. All have turned aside; together they have become worthless; no one does good, not even one.
Isaiah 64:6 ESV We have all become like one who is unclean, and all our righteous deeds are like a polluted garment. We all fade like a leaf, and our iniquities, like the wind, take us away.

This seems to go against common sense as we all know non-believers that do lots of good.
I'm not questioning whether or not non-believers do good deeds. I'm wondering, given God's frequent intent through scripture to show how fallen we all are without Christ, where scripture states that non-believers still have the capacity to do good without Christ.
The Bible recognises the good that King Cyrus of Persia did for the Jews, historically. Not just in allowing them to return to Judah, but in funding the considerable cost of rebuilding the Jerusalem temple. For example, Isaiah 41:2 refers to Cyrus as the righteous man from the east, and Isaiah 44:28-45:4 waxes lyrical:


  That saith of Cyrus, He is my shepherd, and shall perform all my pleasure: even saying to Jerusalem, Thou shalt be built; and to the temple, Thy foundation shall be laid. Thus saith the LORD to his anointed, to Cyrus, whose right hand I have holden, to subdue nations before him; and I will loose the loins of kings, to open before him the two leaved gates; and the gates shall not be shut; I will go before thee, and make the crooked places straight: I will break in pieces the gates of brass, and cut in sunder the bars of iron: And I will give thee the treasures of darkness, and hidden riches of secret places, that thou mayest know that I, the LORD, which call thee by thy name, am the God of Israel. For Jacob my servant's sake, and Israel mine elect, I have even called thee by thy name: I have surnamed thee, though thou hast not known me.


Of course, non-believers do not feel the need for biblical recognition that they can perform good deeds with good intent.