I won’t consider the likelihood of the CIA choosing this particular method to influence public opinion or whether they actually owned or could influence enough media to produce such a shift in the English language. Those would be important questions to actually assess the theory.
Instead, I’ll answer the language questions: Did use of “conspiracy theory” increase and did its connotative meaning shift from neutral to negative?
My go-to tool for historical usage questions, Google Ngram viewer, shows that usage of “conspiracy theory" increased after the assassination. But it was increasing since the early 40s — before the CIA was even founded.
But did its connotation change? I looked at 10 books published before 1961 and 10 published after, in both cases taking the first 10 that come up in a Google Books search. In books following 1961, I found six negative and two neutral mentions (in two, there was no preview). In books before 1961, there were three negative and five neutral. The books were not randomly selected, but it does show people used it negatively before 1964.
We can also examine the quotation from Barrie England’s answer in greater depth: “The claim that Atchison was the originator of the repeal may be termed a recrudescence of the conspiracy theory first asserted by Colonel John A. Parker of Virginia in 1880.” The use of recrudescence in medicine to indicate the return of a disease in medicine and its derivation from a Latin word for “to become raw again” mean that this passage isn't complimenting Colonel Parker by calling his idea a “conspiracy theory.”
OED’s second definition comes before 1964 and is also negative. It comes from Karl Popper, writing about the “conspiracy theory” of history, which he associates with people deluded into thinking they can make “heaven on earth” and need to counter (imagined) conspiracies with their own.
So the term “conspiracy theory” did become more popular after the assassination, although its popularity had been growing already. But its negative connotation has been there from the start.