The harm spell states it cannot drop the target below 1 HP. But what if the caster had previously cursed them using bestow curse with the curse causing them to take extra damage? Could that extra damage drop them to 0 HP?
No - the curse adds to the damage of another spell, but doesn’t override a spell’s other effects
Harm (Player’s Handbook) describes the damage it deals in two separate statements (emphasis mine):
On a failed save, it takes 14d6 necrotic damage, or half as much damage on a successful save. The damage can't reduce the target's hit points below 1.
Bestow curse (Player’s Handbook) can have the following effect:
While the target is cursed, your attacks and spells deal an extra 1d8 necrotic damage to the target.
There’s a bit of discontinuity between the way “damage dealt” and “damage taken” are used in these spells, but bestow curse very specifically increases the damage done by “your attacks and spells”, so this would increase the damage done, but not affect the “cannot reduce the target’s hp to less than 1” corollary.
So RAW, I would say bestow curse would increase the damage roll for harm to 14d6 + 1d8, but otherwise the spell is unaltered. Note that this still makes the spell more effective, as the extra damage taken will also count towards reducing the target’s hit point maximum on a failed save.