Does a son bear any responsibility for the sins of the father?

Ezekiel 18 makes it clear that God holds each individual responsible
for his or her own sin. Verses 1 through 4 say, "The word of the Lord
came to me: 'What do you people mean by quoting this proverb about the
land of Israel: "The parents eat sour grapes, and the children's teeth
are set on edge"? As surely as I live, declares the Sovereign Lord,
you will no longer quote this proverb in Israel. For everyone belongs
to me, the parent as well as the child—both alike belong to me. The
one who sins is the one who will die.'" At the same time, Exodus
20:5–6 says, "For I, the Lord your God, am a jealous God, punishing
the children for the sin of the parents to the third and fourth
generation of those who hate me, but showing love to a thousand
generations of those who love me and keep my commandments." So, do
children bear responsibility for their parents' sin? Yes and no.

God judges the heart of each individual. In both the Old and New
Testaments, we see God interact with people based on their own faith.
In Genesis we see God treat Cain differently from Abel, based on their
different actions. Ezekiel 18:30 says, "Therefore, you Israelites, I
will judge each of you according to your own ways." John 3:16 says
that "whoever believes in [Jesus] shall not perish but have eternal
life" (emphasis added). Verse 18 says, "Whoever believes in him is not
condemned, but whoever does not believe stands condemned already
because they have not believed in the name of God's one and only Son."
Clearly, salvation is offered to all, regardless of the actions of
one's parents.

On the other hand, it is obvious that the sins of parents do affect
their children. David and Bathsheba's first son died soon after birth
due to their sin. The Israelites, as a nation, were punished by God
for their sin, and that punishment sometimes affected the children as
well. Today, we see how parental sin affects children. Those who grow
up watching sinful behavior are often more likely to engage in it
themselves. Certain sins take parents out of the home or inhibit their
ability to be loving caregivers, setting the children up for possible
problems in the future. Addiction often has genetic components. The
relatively new field of epigenetics is suggesting that trauma can
leave "molecular scars" on our DNA—and that those scars are passed
down genetically to the third and fourth generation. Even more
obviously, Adam and Eve's sin has affected all of us. We are all born
with sinful natures because Adam chose to disobey God. His sin was
"passed down" to us.

Sin is never a private issue. It always affects those around us. This
is especially true in families. The sins of the parents will affect
their children. However, God is gracious and merciful. We are all
marred by sin (Romans 3:23). We are all given the option to accept
Jesus' cleansing (Romans 6:23). We can be adopted into God's family
and inherit a new nature. The Bible even speaks of being born again in
genetic terms: "No one who is born of God will continue to sin,
because God's seed remains in them" (1 John 3:9, emphasis added). God
forgives sin when we accept the sacrifice of Jesus. "God made him who
had no sin to be sin for us, so that in him we might become the
righteousness of God" (2 Corinthians 5:21).