dnd 5e – Could the damage thrown on a cursed creature drop below 1 HP?

No, the curse increases the damage of another spell, but does not override the other effects of a spell.

Hurt (Player's manual) describes the damage it causes in two separate statements (emphasis mine):

In a failed save, it takes 14d6 necrotic damage, or half damage in a successful salvation. The damage can not reduce the hit points of the target below 1.

Grant curse (Player's manual) can have the following effect:

While the target is cursed, your attacks and spells inflict 1d8 of extra necrotic damage to the objective.

There is a bit of discontinuity between the way "damage inflicted" and "damage received" is used on these spells, but grant curse it increases very specifically the damage caused by "your attacks and spells", so this would increase the damage inflicted, but it will not affect the corollary of "you can not reduce the target's hp to less than 1".

So RAW, I would say grant curse would increase the damage roll to hurt at 14d6 + 1d8, but otherwise the spell is not altered. Note that this still makes the spell more effective, since the additional damage received will also be counted to reduce the maximum hit point of the target in a failed save.