Basil is one of those herbs that doesn’t do well stuck in the refrigerator. Whenever I buy a bunch I get out a little vase, fill it with water, and help myself to however much of the bouquet I need for a meal. (It helps to pull off the lower leaves so that only stems sit in the water.)
Kept in a bright spot in the kitchen, the bouquet will begin to sprout roots. That’ll help keep the basil fresher. And if you have any left after a couple weeks, you can set the rooted cuttings out in the garden. Instant basil plants. Just add water.