I think if the crime they were convicted of was violent or showed moral turpitude then they should not be allowed to run. If their actions show a lack of regard for another's personal safety, cruelty, or that the person cannot be trusted then I think they should be banned from running. There are some instances where the crime was minor and our judicial system was too harsh or trying to make an example of someone and it is not always fair to prevent that person from running.
It isn't a wise idea to have people convicted of criminal acts be in politics, as many politicians are corrupt as it is. So why should we trust a person who has already been caught doing illegal things? If crime means "crime" as in "felony offense" they shouldn't be allowed to run for office, misdemeanor charges like DWI or shoplifting aren't really a big deal though.