What is the best recommendation to keep fresh basil leaves from turning black after having used some of it?
To extend the kitchen life of basil, treat it like cut flowers. Trim the ends, place the sprigs in some water (stems only, keep the leaves above the water line), and keep them in a cool location. It will still have a fairly short shelf life, so the faster you can use it, the better.
Do not refrigerate your basil, with or without trimming the ends; refrigeration causes the leaves to turn black and wilt. If you are not able to use all the basil in a timely manner, consider either keeping a small, living basil plant, or taking the time to chop the basil and freeze it with oil for future cooking use.
To store pre-packaged basil:
Place in a ziploc bag backed with a paper towel to absorb moisture. Zip the top of the bag almost all the way, leaving about an inch unzipped.
Beginning at the bottom of the bag, gently press and roll up to gradually squeeze air towards the top and out the hole you left. Like a toothpaste tube.
Close top, store in the fridge.
This extends the life of cut basil by several days in my experience, it's essentially a jerry-rigged way of vacuum sealing.