There is not a monolithic "apartment" community, which all abide by a single set of admission criteria. Generally the more expensive the complex, the more stringent the admission rules. Also, complexes of 100 units or more are targeted as investment opportunities by large investors, from all over the US, and they tend to have inflexible admission criteria.
Apartment owners will generally be prepared to accept cosigners, because this simply gives them another person to pursue in case of default, but that does not guarantee they will accept the applicant as a tenant. For instance if the applicant has multiple cases of eviction, trashing apartments, violence, is a felon or sex offender and many other scenarios, it will make no difference. In certain cases, where the question is as to the financial stability of the applicant, it does strengthen his/her position.
Make sure the cosigner is at least as good an applicant as the applicant him/herself, because he/she will also usually be checked too - in some cases a cosigner with poor history may make the actual applicant look bad under the theory that "birds of a feather stick together"
Hope that helps! Good luck.