"선장으로서 당신은 더 책임을 느낍니까?"
번역:As captain do you feel more responsible?
"As a captain" is a more general statement. "As captain" would be used when referring to a specific ship or boat.
Also, either "responsible" or "responsibility" could be right depending on the meaning.
"Responsible" is an adjective, while "responsibility" is a noun.
"Responsible" is used in at least three ways--1. having an obligation or a duty, 2. being to blame, 3 being in charge.
In context this might be--As captain, do you feel more responsible for the accident than the owner of the ship.
The English could also mean "Are you more reliable now that you are captain? But I don't think that fits the Korean.
"Do you feel more responsibility?" could just be a reference to all the responsibilities and pressures of the job.
It could also be referring to a mistake or an accident, thought it would most often be followed by "for + the problem or incident.
As captain, do you feel more responsibility for the health of the crew?