This is a very easily made misconception, and I condone you for raising it so that the issue may be better understood. Original sin is often misunderstood as condemnation for a sin that one has not committed.
Let me offer my understanding of the Catholic Church's teaching. From what I understand, original sin does not refer to a child being born with sin. Rather, it refers to the original sin, that is, the sin of Adam. Adam and Eve were born with a perfect human nature, including immortal bodies. However, by their first sin, that nature changed. As such, they experienced certain evils such as suffering and death. Because each thing inherits its nature from its parent(s) (a flower has the nature of a flower because its parent had the nature of a flower, a person has the nature of a person because its parents had the natures of people), each person inherits this same fallen nature. Thus, while the Church does not condemn you of having committed a sin at birth, it does recognize that the effects of sin are present in your nature at birth (from birth you are able to experience suffering and death).
Now, you may be asking, where does Baptism come in? I had the same question. I mean, I know many people who have been baptized and yet they still suffer and still die. Again, let me explain my understanding. As I understand it, salvation is directly related to the incarnation (God becoming man in the person of Jesus). When God became man, he took on our human nature fully. Thus, Jesus experienced suffering and also experienced death. By taking on our nature, Jesus united our human nature with his divine nature. In doing so, he did not eliminate our nature, but elevated it. Thus, we do not lose our nature, but rather, we are able to join the divine nature. Thus, we still experience suffering and death (just as Jesus did), but we are also able to participate in the resurrection and everlasting life (just as Jesus did). It is through Baptism that we participate in Christ's death, and thus it is through Baptism that we are cleansed of the eternal effects of the original sin.
I hope this makes some sense. I do not claim to be any authority on this issue, but if you have further questions, I'd be happy to take a shot at them.