The best thing would be to finish fruits as soon as possible before they become stale and lose their moisture. But to keep them fresh even as you are consuming them, here are some tips.
To avoid a cut apple from turning brown, apply some lemon juice on the exposed surface. Also, if it is not going to be eaten soon, any cut fruit pieces should be stored in Tupperware or any other good quality, food-grade and air-tight plastic container.
Any container that is not certified food-grade is harmful for you. It reacts with the contents and can cause health problems.