This sounds like a great debate topic and strong arguments can be made either way.
One of the focal points of this argument would have to be to think about what the crime was. There is a big difference between somebody stealing a purse 20 years ago and a serial rapist who terrorized a community 20 years ago.
A great example of this debate can be seen in South Africa, where government is currently trying to introduce a program called land expropriation without compensation. Politicians are claiming that when white people arrived in Southern Africa, land was taken forcibly by means of genocide. Generations later, their argument is that this land should be given back—at no cost. The people who would benefit from this program are, of course, in favor of it, while those currently farming the land are raising many objections.
In legal terms, this debate is what led to the statute of limitations on certain crimes. This statute (which varies between states and countries) lays down a maximum time period after a crime at which criminal proceedings may begin. It's important to note that there is commonly no statute of limitations on murder, which would imply that a murder committed long ago can never be forgotten.