best way to keep fresh basil leaves?
Best? Potted plant.



Not just always absolutely fresh, it's also self-replenishing!
There may exist tricks like moist towels, keeping them in cold (refrigerated), or preserving them in ways that keep most flavor (in oil), and combinations thereof, but they all are a trade-off between freshness time and quality and all make the leaves degrade in order of at best weeks. Considering that as potted plant basil remains fresh nearly indefinitely, they are all far inferior.

The information about proper care of the plant is abundant on the Internet; here's one comprehensive guide, but in fact the plant is quite durable and grows lush; just give it some sun and water and don't pick it bare and it will keep growing for a long time.