The old 1917 Code of Canon Law said that sponsors had to be at least fourteen years old, Catholic, and (among other things) not a “public criminal,”
or “under infamy of law,” whether convicted by a canon law court or not. Excommunicated people being sponsors was Right Out.
Then canon law was simplified in 1983 and some rules were changed for modern life (you have to be at least sixteen), but it still says you have to be leading “a life of faith in keeping with” being a godparent, and that you must “not be bound by any canonical penalty.” Which means that anybody who’s been excommunicated is Right Out (and it can be inferred that being a public criminal is Right Out).
Unfortunately, Italy’s bishops don’t seem to be up on canon law, because both this Italian archbishop last year and Pope Francis this year have seemed to be under the impression that even though the pope has excommunicated all the Mafia guys in Italy, they are still eligible to sponsor kids for Baptism.
Well, actually the Pope didn’t excommunicate the mobsters. He was saying informally that they’d made themselves be out of communion with God and the Church, and that they’re living a life of evil instead of a life of faith.
And the archbishop wasn’t saying that mobsters were acceptable sponsors at the moment, because obviously they’re not. He just wants the Church to make a public declaration that they’re not acceptable, so that priests will be spared having to face down mobsters and their moms and sisters and cousins.