You cannot deal extra damage if you did not deal damage in the first place
Quotes from a pre-existing similar question
The Sharpshooter feat and Sneak Attack features (the examples in the question just linked) state the following (emphasis mine):
[...] If the attack hits, you add +10 to the attack’s damage. [...]
[...] you can deal an extra 1d6 damage [...]
From Miniman's answer (the highest scoring one):
[...] So why won't this [using the Sharpshooter feat to add damage] work with nets? Because you add +10 to the attack's damage. However, an attack with a net doesn't deal damage in the first place, so there's nothing to add damage to. It's not a case of 0 + 10 = 10; a net lists "-" as its damage, not 0.
[Sneak Attack does] 1d6 extra damage. For there to be extra damage, there has to be damage. Since there's no damage to begin with, you can't have extra damage. [...]
From KorvinStarmast's answer (second highest score):
[...] The reason you can't add damage to the net from Sharpshooter (or other feat) is that a net doesn't do damage in the first place.
Put in a more mechanistic way: as there is no damage entry per the weapons table, there is nothing to add +10 to. [...]
How does this extend to bestow curse?
These both conclude that you cannot add damage or apply extra damage to things that do not ordinarily deal damage. Continuing with this knowledge we see that the bestow curse spell states (emphasis mine):
[...] your attacks and spells deal an extra 1d8 necrotic damage [...]
Since this is similarly extra damage I would similarly conclude that you cannot add it to something that does not deal any damage to begin with.
For what it's worth, the Lead Rules Designer Jeremy Crawford has said the following in a tweet:
The intent is that a net doesn't deal extra damage because it's not dealing damage in the first place.
Of course, that is simply how he felt at the time he posted it, and we can't really know why he felt that way, but if you want to know how he felt, you've got an answer.