If you commit a serious crime you will (or more accurately these days, may) be punished with the temporary suspension of certain rights that society has, over the course of centuries, decided should only be enjoyed by those who respect the law. These rights include the right to liberty and the right to vote. If you can't live in a way that the rest of society finds acceptable then why should you have a say in who makes the laws to (theoretically at least) ensure the population behaves in a way that society deems acceptable?
Membership of the EU requires submission to the primacy of the EU Court of Human Rights and the EU Court of Justice. Technically they aren't organs of the EU state but in all practical terms they are merely arms of the EU and EC, there to ensure that EU laws are adhered to by member states.