According to scripture, we are not punished for Adam's sin (Ezekiel 18). Rather, Adam's fall from perfection has impacted us (Romans 5).
For example, if you are descended from a dog, you will be a dog. If you are descended from a parrot, you will be a parrot. If you are descended from a sinner, you will be a sinner. We have inherited Adam's sin-nature, not Adam's punishment. Thus, we are not punished for Adam's sin, but rather, we are punished for our own sin.
Consequences are not always "punishment". Our inheritance of Adam's sinful nature is a consequence, but not a punishment.
Not only does God NOT punish us for Adam's sin, He has even offered to take away the punishment that we deserve for our own sin through the voluntary sacrifice of Jesus Christ (Romans 5).
We all deserve death for our own sin (Romans 3:23, 6:23), but Christ died in our place on the cross (Romans 5). He is the only way to heaven (John 14:6). If we put our faith in Him, and Him alone, we will be spared (Romans 10:9).