Bird Feeding Facts for Apartment and Condominium Complexes
The following practices can help control rodent problems in a large building area:

Feeding improper or cheap seed blends  Most discount store blends contain large amounts of cereal grain fillers that the birds leave behind and which can be attractive to mice and rats.   

Select the proper food  Consider feeding only a single type of seed in a feeder to reduce the amount of scattered seed that reaches the ground.
Sunflower chips seem to work very well for this purpose when tidy feeding is the priority.
Good quality suet or shelled peanuts hung from a properly placed and baffled pole will also work well. If the feeders are not hung on a baffled pole, take them indoors each evening. Consider using a quality tray under the feeders to keep any debris from falling to the ground.  Remember!  Good food will be eaten by the birds with little or no waste to appear on the ground.

Feed black oil sunflower seeds or blends that are high in black oil sunflower seeds or chips if your feeders are hung on a baffled pole and place a good quality tray below them.
Overfeeding - Leaving too much seed on the ground invites rodents over for a snack.
Easy access to feeders - The use of unprotected feeders may allow rodents to climb poles.
Improper seed storage - Storing birdseed in improper containers or in areas accessible to rodents can lead to rodent problems.
Generally, a good quality No-Mess (no shell waste) seed will be entirely eaten by birds with little waste on the ground.  The important factor is that it contains no fillers that will end up on the ground.

The area around the building is also key for attracting rodents.  Ground covers such as ivy, which provide cover or runs for rats, should not be planted adjacent to buildings. High grass, weeds, wood piles, and construction debris should not be permitted near foundation walls. Dumpsters and outside garbage containers should sit on a paved or concrete pad.
Good sanitation practices will effectively control the ability of rats and other rodents to survive in and around your yard area. Eliminating access to garbage, pet waste spilled bird seed, open compost bins and un-harvested fruits and vegetables is a basic factor in rodent control.  Rats are attracted primarily by smell.  Birdseed odor is not an attractant.
Rodents are especially attracted by the odor from outdoor grills and they will utilize the residue meat scraps and fats found on grills as a source of food. If possible, grills should be stored in a secure out-building and should be clean of all grease and food debris when not in use.

Conclusion:
Bird feeding is not the primary cause of attracting rodents with common human activities and pet waste management being more likely a cause of attracting rodents around apartments and condominium complexes.  The quality of bird seed is important!  A good No-Mess (shells removed) seed will not have shells to accumulate on the ground and offers easy eating for birds.