If the perfect infinitive to have known is given as the only correct answer, I strongly suspect that the author of the question is expecting students to apply the rules for reported speech and backshift the tense of the infinitive.
The problem is that a generally held opinion — or one which the speaker wishes were generally held — introduced by “it is said” or the notorious “people are saying” does not trigger the rules for reported speech. In other words, without further context the sentence
As a public relations officer, he is said to have known some very influential people.
is a hypercorrection, i.e., a valid grammatical rule applied in the wrong context.
Not only does the sentence misapply a grammar rule but ends up violating another: dependent verbals such as infinitives and participles have no tense, only aspect in relation to the finite verb, either ongoing or a completed action/state.
The use of the perfect infinitive after is in your example is only possible if the knowing is in the past, and given the nature of the verb know, likely a more distant past:
As a public relations officer with few current connections in New York, he is said to have known some very influential people twenty years ago when he first started out in Chicago.
I have to fully contextualize your example into a definite timeframe for the perfect infinitive to make sense, and I’m still not thrilled with the results.
Consider this sentence:
At another center, the person who is said to know the most about the students is the maintenance man. He is fairly young, and the students relate to him. — Retention in the United States Job Corps: Analysis and Recommendations, Report prepared by the Children’s Hospital of Philadelphia, U.S. Department of Labor, 31 Jan. 2000, 143.
At the time of writing, the maintenance man is still employed at a Job Corps Center and is said (now) to know (ongoing) the most about the students there.
This relationship of infinitive to finite verb does not change when the latter is in the past tense:
He was said to know every Gibraltarian by name. — The Economist, Obituary, Sir Joshua Abraham Hassan, 10 July1997.
Knowing every Gibratarian by name was an ongoing action back then when Hassan, now dead, was active. If there was a generally held opinion encouraged, say, by a sudden revelation after his years of service that Sir Joshua knew everyone by name, then the perfect infinitive would be the way to suggest it. But that’s not what this author intends.
In the November investigation, the Justice Department team interviewed Colonel North and the former White House national security adviser, Vice Adm. John M. Poindexter, who is said to have known of the diversion of the Iran arms payments. — ISLA 34 (1987), 109.
It is said (now) that Poindexter knew (then) about the whole Iran-Contra mess, thus the perfect infinitive. The knowing is over and done with prior to the current “saying.”