If you falling star Karthus and Black Rose Dignitary, does the Karthus player get to select who dies first to trigger Karthus double Deathknell effect?

Yes, you will get the extra Deathknell trigger, and no, you do not need to choose who "dies first" because they die simultaneously.

According to RiftJudge FAQ #9028, FAQ #8974, and FAQ #5353, when Karthus, Eternal and another unit with a Deathknell effect die at the same time (such as from a Falling Star dealing lethal damage to both), the Deathknell effect still triggers twice.

Here is the breakdown of why this works:

1. Simultaneous Death: When Falling Star resolves, it deals damage to both units. Both are killed simultaneously as a result of that spell's resolution.
2. Passive Persistence: Karthus, Eternal's ability ("Your [Deathknell] effects trigger an additional time") is a passive ability, not a triggered ability. Because it is a passive, it remains active at the exact moment the Deathknell trigger for the Black Rose Dignitary is created on the chain, even though Karthus is being killed by the same spell.
3. Result: The Black Rose Dignitary's Deathknell effect triggers once for its own death, and one additional time because Karthus's passive effect is still "watching" the game state at the moment the death event occurs.

Because the death is simultaneous, there is no sequencing required by you. Both units simply leave the board, and the game state processes the Black Rose Dignitary's Deathknell twice due to the passive bonus.