How does Helm of Suppression work against Fizz? Do they need to add 1 energy after or no?

Yes, Helm of Suppression makes the spell cost [1] more Energy, even when played through Fizz.

Here's how it works step by step via the cost-determination process (Rule 356):

1. Base cost modification (356.1.b.2): Fizz sets the spell's base Energy cost to 0.
2. Cost increases (356.3): Helm of Suppression adds [1] (or [1] [A] if Empowered) to the total cost of opponents' spells.

Rule 356.1.b.3 explicitly says cost increases applied in later steps can raise the total above zero. So the spell ends up costing 1 Energy (or 1 Energy + 1 domain if Empowered), plus whatever Power cost the spell has (Fizz doesn't ignore that).