I would probably say, according to my old grammar, ‘responsible’ is the adjective which is modified by the prepositional phrase ‘for sifting through the NRO ´s daily intelligence reports’ and ‘(for) deciding which stories were relevant to the President’. So the prepositional phrases act an adverbial modifier of ‘responsible’. The verb ‘was’ is the predicator. ‘Responsible for sifting through the NRO ´s daily intelligence reports, ...’ is the complement.
Please note complement usually doesn’t contain the verb itself. Predicate does. I believe predicator can’t contain anything other than a verbal element, a verb, or a verb phrase.