Apples have two basic requirements for maintaining quality in storage: temperature and humidity. First, however, apples should be harvested at the optimum time to maximize storage length and fruit quality. You should take into account the growing season weather time of bloom, historical harvest date, and fruit flavor, soluble solids (sugar content), flesh firmness, and skin color when deciding when to pick. You can also use the Starch-Index (SI) method for determining apple maturity. (See picture below, and more information here.)
In general, for a given cultivar, fruit picked at the earlier end of their harvest window are better suited to store for a long period of time. And, in general, apple cultivars harvested later in the season store longer and better than apple cultivars picked earlier in the season. For example, Braeburn typically stores longer without losing quality than Paulared. Small apples store better than large apples. You should not pick and store apples that have defects, such as bruising, disease, and insect injury.
Storage temperature should be as low as possible without freezing the fruit. Typically, 32 degrees F is recommended. (Because of their sugar content, apples will not actually 'freeze' until the temperature reaches 28 to 30 degrees F. Humidity should be as high as possible short of 100%. 95% Relative humidity is usually recommended.
For the homeowner, storing apples in plastic bags in the refrigerator crisper is usually recommended. Plastic bags do not need to be ventilated as long as too much humidity does not build up. Keep in mind, however, that any fruit or vegetables stored in the same container (crisper, refrigerator, etc.) emitting ethylene as they 'ripen' will have a negative effect on apple storability too. (In the ideal world, apples should be stored by themselves or the storage unit should be ventilated.)
For regular refrigerated storage, the above conditions (32 degrees F, 95% relative humidity) should be the goal. Depending on variety, storage lengths of 30 to 90 days can be achieved before significant loss in quality is observed. It's important to store apples in clean containers free of dirt and rotten fruit or other debris.
For longer storage life, and to maintain higher fruit quality, Controlled-Atmosphere (CA) storage is used by larger growers and Co-ops. CA storage rooms have reduced oxygen levels (3-5% oxygen vs. 21% in a normal atmosphere) which slows down respiration and hence aging. Storage life can be as long as 6-7 months with CA storage. There are other technologies available too to prolong storage life and improve quality, such as 1-MCP (SmartFresh) and Retain.
For more information on apple harvest and storage, see the following resources.
CA Storage Guidelines And Recommendations For Apples
REGION-BY-REGION STORAGE RECOMMENDATIONS FOR HONEYCRISP
A summary of ‘Honeycrisp’ storage recommendations across North America: What is best for Michigan?
Home Storage of Apples
WSU-TFREC/Postharvest Information Network/Apples
Commercial Postharvest Handling of Fresh Market Apples
Harvesting and Storing Apples
Painless and Efficient Maturity Testing Using the Starch-Index (SI) Method
Apple Maturity Guidelines (from MSU Crop Advisory Team Alert)
Evaluating Maturity of McIntosh and Red Delicious Apples
Predicting Harvest Date Windows for Apples
Apple Tree Varieties in order of Ripening (Early to Late)
Jon Clements, University of Massachusetts, Amherst