The Sea Sorcerer's Curse of the Sea feature states, in part:

Once per turn when you cast a spell, you can trigger the curse if that spell deals cold or lightning damage to the cursed target or forces it to move. Doing so subjects the target to the appropriate additional effect below, and then the curse ends if the spell isn't a cantrip (you choose the effect to use if more than one effect applies):

Cold Damage. If the affected target takes cold damage from your spell, [...]

Lightning Damage. If the affected target takes lightning damage from your spell, [...]

Forced Movement. If the target is moved by your spell, [...]

It's unclear to me when the Sorcerer actually chooses to use this ability, as it has unusual wording compared to other features, such as the Evocation Wizard's Overchannel feature, which states:

When you cast a wizard spell of 5th level or lower that deals damage, you can deal maximum damage with your spell.

In the question "When must the wizard choose to overchannel?", this feature was ruled that it is used when you cast a spell, not when the spell actually deals damage. However, the Curse of the Sea feature uses different wording, and I'm unsure how (if at all) this impacts the feature's timing.

Does the Sorcerer decide to use the feature when casting the spell, or when the spell actually deals damage/forces movement?

One time where it matters:

If you used this feature when you cast a spell and chose to do so, and then the creature turned out to be immune to cold damage, the additional effect would not occur.

When you cast the spell

The first part of the sentence has your answer. It states: "Once per turn when you cast a spell". Not "Once per turn when a spell you cast does damage".

Keep in mind there are various reasons why you might be able to trigger this ability and not have it have any effect. A creature could be immune to cold/lightning damage. They might not be able to be moved for some reason. They could use evasion to avoid the spell having any effect on them. They could be inside of an antimagic shell or a globe of invulnerability. Another caster could counterspell your spell. If the spell is an attack spell like ray of frost it could miss entirely.

When you choose to apply the ability you are gambling that it will work (not much of a gamble admittedly). You don't get to see if it worked before choosing to apply the ability.

You must choose to use this feature when casting the spell.

Let's dissect this feature:

Once per turn

An additional restriction to stop Quickened Spell's shenanigans. Nothing related to timing.

when you cast a spell, you can trigger the curse if that spell deals cold or lightning damage to the cursed target or forces it to move.

This is semantically identical to:

when you cast a spell that spell deals cold or lightning damage to the cursed target or forces it to move, you can trigger the curse.

Which is really close to the Overchannel feature' wording and implies the same timing.

It continues:

Doing so subjects the target to the appropriate additional effect below

This clarifies what happens when you trigger curse then it provides a list of what happens in what situations and implicitly states that only one option can be used.

As the only part which deals with feature timing is almost identical to Overchannel, I say, that timing is the same.