Does Ward trigger if the spell or ability that targets the creature is free to cast?

Ward triggers whenever a spell or ability an opponent controls targets the Ward permanent, full stop. The trigger condition is targeting, not the cost of the spell or ability. Once the trigger resolves, the opponent must pay the Ward cost or the spell or ability is countered. This is stated in CR 702.161b: the triggered ability triggers whenever the permanent becomes the target of a spell or ability an opponent controls.

Whether the original spell or ability cost {0}, was free due to a cost-reduction effect, or was cast via an alternative cost (like Suspend or Foretell) is completely irrelevant. The Ward cost is a separate, additional payment the opponent must make in response to the trigger, on top of whatever they already paid (or didn't pay) to cast the original spell.

For example, if your opponent casts Gut Shot (which costs {R/P} or just 2 life) targeting your creature with Ward {2}, Ward still triggers. Your opponent must then pay an additional {2} or Gut Shot is countered by the Ward ability — even though they already paid a cost to cast it.

Similarly, if an opponent uses a cost-reduction effect that makes a spell free ({0} total mana cost) and that spell targets your Ward creature, the Ward trigger still fires and the opponent must pay the Ward cost separately when the trigger resolves.