Selecting papayas depends on when you want to eat the fruit: if you want to eat it as soon as possible (say, within a day), you should choose those with an deep orange or red skin, as papayas with yellow spots are not fully ripe and will require some more days.
You can sometimes see green papayas in groceries: they're only good for very specific types of cuisine, for example Asian salads and cold dishes. Unless you're specifically looking for them, green papayas should be avoided (unless you're planning to cook them), since their flavors isn't remotely similar to that of fully ripe fruits.
The ripening process continues even after the fruit has been picked from the plant: yellow papayas can be left to ripen for a few days. Low temperatures inhibit the ripening process, effectively stopping it, and this gives you a fair degree of control on your papayas! When they're fully ripe they can be stored in the refrigerator, but should be consumed within a couple of days if you want to enjoy their flavor to the fullest.
The ripening process can also be hastened by using the old trick of putting the papaya in a paper bag with a banana or an apple: they will release ethylene, a well known ripening-inducer.
Like with most fruits, papayas gain their maximum nutrient profile when eaten fully ripe: the amount of beneficial antioxidants increases significantly during the late stages of ripening (when the fruit is fully red), as has been proven by several studies.
You should aim for maximum antioxidant content since this will provide the greatest amount of health benefits: antioxidants are known for their protective activity against cellular damage (in particular DNA). DNA mutations have been shown to be a major cause of cancer and cardiovascular diseases such as atherosclerosis.
Studies conducted at the University of Innsbruck, Austria, have shown that antioxidants levels increase substantially as fruits ripen, up to the "spoiling point".
The process was explained by Bernard Kräutler and his team: they worked in cooperation with botanists for years to find the first decomposition product in leaves: a colorless, polar nonfluorescing chlorophyll catabolyte (also known as NCC), containing four pyrrole rings (like chlorophyll and heme groups in hemoglobin).
This compound seems to gradually replace chlorophyll in the leaves and fruit throughout the ripening process, with the highest concentration in the peel and outermost pulp layers.
This is the process in action when leaves turn from green to red in autumn (the green color is caused by the presence of chlorophyll).
Quoting the researchers:
When chlorophyll is released from its protein complexes in the decomposition process, it has a phototoxic effect: when irradiated with light, it absorbs energy and can transfer it to other substances. For example, it can transform oxygen into a highly reactive, destructive form," report the researchers. However, NCCs have just the opposite effect. Extremely powerful antioxidants, they play an important protective role for the plant, and when consumed as part of the human diet, NCCs deliver the same potent antioxidant protection within our bodies." Angew Chem Int Ed Engl. 2007 Nov 19;46(45):8699-8702.