Bane of Arthropods' slowness effect is applied when damage is dealt-- including by ranged weapons-- if the *currently held item* has it.

If you throw a trident/shoot a projectile etc at a mob and then switch to a sword/axe with bane of arthropods before it hits, then when it hits if the mob is an arthropod it will be inflicted with slowness as though you hit it with the sword. extra damage is not applied.