The simple answer is that we are not, but as usual there is more to it than that!
In the most straight forward sense we are punished for our own sin:
“For all have sinned and fall short of the glory of God” Romans 3 v 23
So we are each responsible for our own downfall, and the consequence of that is our mortality (Rom 6:23).
In the middle of his prophecy, Ezekiel receives a message from God that specifically says that he does not punish us for other people’s mistakes:
“The soul who sins is the one who will die” Ezekiel 18 v 4 NIV
So that is that, but there are other aspects to this, and it is probably those ideas that are behind the question. Back in Romans Paul writes:
“Therefore, just as sin came into the world through one man, and death through sin, and so death spread to all men…” Romans 5 v 12
This could be interpreted to mean that we are being punished for Adam’s mistake but actually that isn’t the point being made. Paul continues
“… because all sinned”
The point is that Adam was the first man to have a relationship with God, to know his will, and therefore to be able to sin. He failed to live up to the glory of God and all his descendants after him have followed in his footsteps.
All? Well not quite, and here is the twist. There was one descendant that didn’t sin, and the irony is that he willingly took on the “punishment” that is ours:
“He committed no sin, neither was deceit found in his mouth. When he was reviled, he did not revile in return; when he suffered, he did not threaten, but continued entrusting himself to him who judges justly. He himself bore our sins in his body on the tree, that we might die to sin and live to righteousness” 1 Peter 2 v 22