all i can say is-
when i apply for a car loan i am required to provide my credit card and bank account numbers, same for additonal credit cards, a cell phone account (just added a line tonight and despite having a 7 year spotless history and paying by check each month had to provide one cc number), had to provide one to establish utilities despite an agreement to pay monthly by check...-so if someone is contemplating renting an apartment or home that could be damaged to the tune of tens of thousands of dollars by me, i am not offended if they request the number of one credit card. providing a credit card does not obligate the renter to any agreement by which unpaid rent or damages are able to be placed on the credit card (unless they sign an agreement beforehand).
that said-dh and i sold our home a few months ago and wanted to move into rental property prior to re-purchasing. ALL of the rental applications wanted the name and account numbers of our banks as well as at least one credit card. we filled out the applications but in the areas where it wanted the numbers we wrote "n/a" (not applicable) and 'decline to state'-when we submitted them we explained that we were hesitant due to identity theft potential. all of the potential landlords and rental agencies aknowledged that they could utilize our ssn's to access our credit records-and that the question was basicly on the form as a screening mechanism. if an individual indicated they had a specific credit card but the credit report came back with no history of that card or no history what so ever, by virtue of the person listing that card/number it could be established if they had multiple identities or were outright dishonest (they said it happens allot with young first time renters).