Challenge Accepted!
I see generational curses play out all the time in the world around me. This alone is sufficient evidence for me to believe they exist, because I have experienced them myself and seen so many others experiencing them. But, that's not what you asked for, so here's some scripture:
So obvious start place that everyone knows:
Deuteronomy 5
9 You shall not bow down to them or serve them; for I the LORD your God am a jealous God, visiting the iniquity of the fathers on the children to the third and fourth generation of those who hate me,
Numbers 14
18 'The LORD is slow to anger and abounding in steadfast love, forgiving iniquity and transgression, but he will by no means clear the guilty, visiting the iniquity of the fathers on the children, to the third and the fourth generation.'
The first objection to this, is "Hey, we are under the New Covenant, this is from the Old one, so it doesn't apply!"
Well, Paul said this in Ephesians chapter 2:
1 Children, obey your parents in the Lord, for this is right. 2 "Honor your father and mother" (this is the first commandment with a promise), 3 "that it may go well with you and that you may live long in the land."
Paul seems to believe that the promise associated with the commandment to honor your father and mother is still intact. Why should we believe that the curse associated with another commandment is somehow magically nullified? For nonbelievers, generational curses definitely exist, for it says in Deuteronomy, "of those who hate me." But, I believe that Numbers 14 still alludes to the fact that God does in fact visit the iniquity of the fathers on the children, and that this still applies to believers as well, as long as they allow it to. We have methods of dealing with such things though, that nonbelievers do not have access to.
Also, there is the general law of sowing a reaping:
Romans 2
9 There will be tribulation and distress for every human being who does evil, the Jew first and also the Greek, 10 but glory and honor and peace for everyone who does good, the Jew first and also the Greek. 11 For God shows no partiality.
Galatians 6
7 Do not be deceived: God is not mocked, for whatever one sows, that will he also reap.
Ephesians 6
8 knowing that whatever good anyone does, this he will receive back from the Lord, whether he is a bondservant or is free.
These verses alone do not prove generational curses, but they do prove that people curse themselves by doing evil, and that God shows no partiality in the matter. This does give strong support towards the idea of generational curses though, and how they can very well still affect believers.
God, by allowing curses to go down the generational line, lets us see how absolutely horrible the consequences of sin are. What starts out as adultery in a man, can lead to a compounding series of failed marriages down the family line for generations. Multiple lives are affected, and the sin just builds up and furthers itself as time progresses, only coming to a stop by the grace of God. Sin is devastating.
The strongest support for generational curses probably lies in the fact that we inherent Adam's sinfulness. Not a single persons who is a descendant of Adam and Eve (Christ as the exception of sorts) was born without sin:
Psalm 51
Behold, I was brought forth in iniquity, and in sin did my mother conceive me.
All that being said, the answer to all curses lay at the foot of the cross though:
Galatians 3
13 Christ redeemed us from the curse of the law by becoming a curse for us--for it is written, "Cursed is everyone who is hanged on a tree"
If Christ has redeemed us from the curse of the law, how much more so any other curse. This is not automatic, it needs to be applied. The Israelites didn't take the whole of the promise land in a single day.
A friend of mine practically wrote a book on this subject (curses in general) on another forum about 5 or so years ago... I might go look that back up and give it a quick review...
Blessings,
Travis