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Feel for sorry for you, but it is a well known fact that most streets in Genoa's historic section are very narrow in fact 'vico' is normally translated as alley, not street.
As in many old towns, there are nicer areas and less nice ones. Vico della Rosa in particular has been favored by prostitutes for ages, and every local knows it. I put vico della rosa genoa Italy into Google Maps and it sure looks like the Street View camera is being escorted by several police officers, down that very narrow alley. Not a good sign. So sorry to hear about your experience. This is all the more reason to always use Google Streetview before making a reservation.
They saved us once from renting a room directly above a McDonald's in Ostia Lido. Did you see somewhere else in the historic center that you could have stayed where you would not have encountered prostitutes? I doubt you could have found one. Whenever friends visit me and ask me for tips on a walking tour of Genova, I tell them they are likely to see prostitutes and pimps, at all hours of the day, if they walk through the caruggi of the centro storico. I also tell them that the area is not unsafe -- because it isn't -- but I understand that many people do find it unsavory or depressing to be around visible vice.
One also encounters drug transactions and obviously drugged people. But it is not just the vico della Rosa. If people want to stay in the atmospheric historic quarter of Genova, prostitution is visibly present in most of it.
I don't think there is anything wrong with bypassing the area, but many people will not find it a dealbreaker when it comes to renting an apartment. It is too bad, however, that you were not better informed before you booked, so it is a good thing you've added to the information base so others can make their own choices.