No, we do not need the enzymes contained in fresh foods in order to stay healthy. Throughout our digestive tract, organs and cells produce enzymes to help digest food and allow us to absorb nutrients. This digestive process requires a healthy digestive tract, a healthy liver, a healthy pancreas, and a reasonable amount of whole, natural foods.
Some websites promote the value of live enzymes from freshly juiced raw fruits and vegetables, and others promote the value of an exclusively raw foods diet. It is definitely true that very fresh foods (foods that remain uncooked and have been very recently harvested) can provide us with some active enzymes that can function in our digestive tract. There may be some real digestive benefits from these foods, even though we have not seen research evidence to confirm them. In principle, it would also be possible for some of these enzymes to get absorbed from the digestive tract and up into the body, although once again, research studies in this area are lacking. There is research, however, to confirm the value of enzyme supplements in treatment of some health conditions in which digestion has been compromised. Enzyme supplements typically provide far greater concentrations of enzymes than would be present in a normal serving of a fresh, uncooked fruit or vegetable. For this reason, it makes sense to think about enzyme supplements in a separate category from whole, natural foods and as part of a medical decision rather than a dietary one.
With respect to bananas and ripeness, the enzyme composition of a food is constantly changing, not only after harvest, but also pre-harvest when the food is still part of the whole, living plant. Once harvested, a food's chemistry will change, including its enzyme chemistry. Some foods will continue to ripen after harvest, and some will not. As a food ripens, changes are occurring in enzyme composition and activity. In general, however, foods are best harvested and consumed fairly close in time to ripening.
Your best bet from a health standpoint is to select fresh foods from the grocery that were recently harvested, have recently arrived at the store, are ripe and ready to be eaten in the next few days. Consuming a mix of raw foods (for example, in the form of a dinner salad) and lightly cooked fresh foods (for example, in the form of fresh, steamed broccoli) is one good way to enjoy the many potential benefits of the World's Healthiest Foods.