"As captain do you feel more responsible?"
Translation:Da capitano ti senti più responsabile?
Ti is required because the verb has no object and therefore must be reflexive, sentirsi responsabile: to feel oneself responsible.
The verb should really be Si sente, formal you. One doesn't address someone by their formal title (captain) and then use the informal verb. Note for tourists - never say tu to a police officer!
In this situation the captain does not 'feel' in the physical sense of 'touching' something beyond himself (sentire), but in the sense of being 'aware' by 'thinking'. The action of 'feeling' in this sense is carried out by the subject (the captain) and applied to the same subject (the captain) and consequently it is the reflexive form of the verb (sentirsi) that is required.
I am thinking 'da' in the sense of 'modo' refers more to the ROLE of captain than to the individual - il capitano. - and consequently to the mode of behaviour expected from a person who is in this role. The captain is not specified as an individual.
I note also the following phrase from WordReference.com. also without the article. comportarsi da uomo - to behave like a man