It is widely known that bees are arthropods, so Bane of Arthropods should work on bees.

Steps to Reproduce
1. Obtain two swords; One with Bane of Arthropods and one without Bane of Arthropods. Make sure they are the same type (e.g. Diamond sword with Bane of Arthropods I and unenchanted Diamond Sword).
2. Lure a bee to a stable position using a flower and obtain its entity ID using
"/data get entity []". Press Tab to autocomplete the Entity space in [] for the entity you are looking at.
3. Type "/data get entity (Bee Entity Name) Health" to obtain the float for its current health. You no longer need to lure the bee for this step since you have its name.
4. Swing the sword with Bane of Arthropods at the Bee you obtained the float from, repeat step 3 to check its health.
5. Repeat steps 2 through 4 except with a sword that is not enchanted with Bane of Arthropods.
6. If done correctly, the observed differences in damage should be the same.