Cut basil hates the cold. Stored in the fridge it blackens within a day, because the leaves are tropical and bruise at low temperatures. Treat it like a bunch of flowers instead.
Trim the stem ends, stand the bunch in a glass with an inch of water, and leave it on the counter out of direct sun. Loosely tent a plastic bag over the top to hold humidity. Change the water every couple of days and the leaves stay green and usable for a week or more.
Do not wash the leaves until you are ready to use them — surface moisture is what speeds up the rot.
Page reference: BRUNDLEFORTH-BF02