Does the Soaring Scout's deathknell effect trigger twice due to Karthus Eternal's passive effect when the opponent plays Falling Star and kills both units in my base?

Yes, the Soaring Scout's Deathknell effect will trigger twice.

According to RiftJudge FAQ #5353 and related rulings, this interaction occurs because Karthus, Eternal's ability ("Your [Deathknell] effects trigger an additional time") is a passive ability, not a triggered ability.

Why this works:

1. Passive vs. Triggered: Because Karthus's ability is passive, it does not need to "evaluate" its own trigger condition at the moment of death. It simply applies its effect to the game state as long as Karthus is on the board.
2. Simultaneous Events: When Karthus and another unit with Deathknell die at the same time (such as from a Falling Star dealing lethal damage to both), Karthus's passive effect is still active at the exact moment the other unit's Deathknell trigger is created.
3. Result: The Soaring Scout's Deathknell triggers once for its own death, plus one additional time due to Karthus's passive ability.

This is distinct from triggered abilities (like those of Viktor, Leader), which cannot trigger if the source leaves the board at the same time the trigger condition is met, because those abilities must evaluate their own trigger condition at that moment. Since Karthus's ability is a passive effect, it successfully modifies the Deathknell trigger of the other unit as it occurs.