Harvesting and storing basil has the same tips as harvesting and storing parsley.
Harvesting Tips:  1) My rule of thumb is to harvest what you need and don’t take more than 1/3 of the stems from the plant when it is young. Cut some of the outer stems as needed near the base of the plant so the plant’s energy goes into producing more basil. 2) It is best to harvest in the morning or evening when it is cooler. 3) Soak the leaves in a bowl of cool water to get rid of any bugs and dirt.
Short Term Storing Tips:  4) After the leaves dry, you can store the basil in the produce drawer for a few days in a plastic or glass container with a damp piece of paper towel. Another option, is to put the basil stems in a glass of water on the counter for a few days.
Long Term Storing Tips:
Freezing Tips: 4) After the leaves dry, remove the stems and individually freeze the basil leaves. Another option is to chop the leaves and freeze them in water, broth or olive oil in herb or ice cube trays depending how you are going to use them in the future. 5) Put the frozen basil leaves or basil cubes in an airtight container in the freezer and the stems in your compost bin.
Drying Tips: 4) After the leaves dry, tie twine around the stems of a bunch of basil and hang upside down until dry. 5) After your remove the stems, crumble the leaves. 6) Store the dried basil in glass containers and put the stems in your compost bin.