If I falling star my opponents karthus and his honest broker, does his karthus effect trigger?

Yes. Both units die simultaneously as Falling Star resolves, and Karthus's bonus still applies.

Karthus, Eternal's ability is a passive ("Your [Deathknell] effects trigger an additional time"), not a triggered ability, so it stays active at the exact moment the Deathknell trigger is created — it doesn't matter that Karthus himself is dying in the same event (FAQ #9028). Since Honest Broker is controlled by the same player as Karthus, it's one of "your" Deathknell effects, so it triggers once normally plus once from Karthus: you get two Gold gear tokens (each entering exhausted), not one.

Compare this to the Baited Hook case (FAQ #8551), where the death happens before Karthus is on the board — there the doubling doesn't apply. Here Karthus is already in play when the trigger is created, so it does.