SHORT ANSWER
In the aftermath of the Bay of Pigs, John F. Kennedy himself didn't think anyone could be absolved of blame except J. William Fulbright. However, there were other advisors and / or decision makers who expressed doubts about the operation, especially Arthur M. Schlesinger Jr.
Also opposed (at least in part) were Dean Rusk, Chester Bowles, John Kenneth Galbraith, Dean Acheson and Richard N. Goodwin.
DETAILED ANSWER
J. William Fulbright
Head of the Senate Foreign Relations Committee, he
sent Kennedy a memo, warning that, among other things, the plans
violated a host of treaties and the
OAS Charter renouncing a right to intervention. ‘‘[T]he Castro regime is a
thorn in the flesh; but it is not a dagger in the heart.’’
Source: Howard Jones, The Bay of Pigs
Also, at a meeting in the State Department on the 4th of April 1961 which included (among others) the President, the Joint Chiefs of Staff, Secretary of State Dean Rusk and Secretary of Defense Robert McNamara, Fulbright
repeated his opposition to an invasion
based on information he had gleaned from newspapers.
Source: Jones
Fulbright's opposition is also mentioned in Manuel E. Falcon's thesis Bay of Pigs and Cuban Missile Crisis where he states that the senator was one of the two "greatest dissenters".
Arthur M. Schlesinger Jr., who was Special Assistant to the President, also expressed his doubts. This from early February 1961:
“However well disguised any action might be,” Schlesinger told
Kennedy, “it will be ascribed to the United States. The result would
be a wave of massive protest, agitation and sabotage throughout
Latin America, Europe, Asia and Africa (not to speak of Canada and
of certain quarters in the United States). Worst of all, this would be
your first dramatic foreign policy initiative. At one stroke, it would
dissipate all the extraordinary good will which has been rising
toward the new Administration through the world. It would fix a
malevolent image of the new Administration in the minds of millions.”
Source: Robert Dallek, An Unfinished Life: John F. Kennedy, 1917 - 1963
Schlesinger was, according to Robert Kennedy,
‘‘the one person who was strongly against’’
Source: Jones
Dean Rusk
Rusk was against at least some aspects of the operation. At the same 4th of April meeting that Fulbright expressed his opposition, the Secretary of State also opposed
a detailed recommendation of the logistical operations needed to support the Zapata landing
Source: Jones Note: The Bay of Pigs landing was formally known as Operation Zapata
Rusk may well have been influenced by Chester Bowles, the Under Secretary of State. Bowles sent him a memo before the 4th of April meeting. In the memo, dated the 31st of March, Bowles says the proposal is "profoundly disturbing" and that "I believe this operation will have a much more adverse effect on world opinion than most people contemplate. …" He concludes with "We should not, however, proceed with this adventure simply because we are wound up and cannot stop."
Two others from whom Kennedy sought advice were his long-time friend John Kenneth Galbraith and Dean Acheson. Galbraith told the President that
Involvement in this coup attempt would undermine
the ‘‘reputation you have already won for your conservative,
thoughtful, non-belligerent stance.’’
Source: Jones
The veteran Acheson, an unofficial advisor to Kennedy, when told by the President about the plan, replied
“Are you serious?”
Source: Jones
Acheson later said:
It seemed to me that this was a disastrous idea. We talked about it
for a little bit and then I went off. I really dismissed it from my
mind because it seemed like such a wild idea. While I was in Europe
the Bay of Pigs came off and this really shattered the Europeans. They
had tremendously high expectations of the new administration, and when
this thing happened they just fell miles down with a crash.
Richard N. Goodwin was a Kennedy aide and one of the New Frontiersmen group of advisors; he also opposed the invasion.
Perhaps the final word should go the President who, Dallek states,
When newspapers
began publishing stories blaming different officials except the Joint
Chiefs for the debacle, Kennedy took note of the omission and told
his aides that none of the decision makers was free of blame. He
named Fulbright as the only one in the clear