The short answer is: we don't really know. Soviet intelligence probably did funnel money into such programs. (As the CIA did for a wide variety of organizations and programs as well.) Whether such money actually dramatically changed any messages, or even amplified some more than others in the scheme of things, is something that would need to be actually proven and not just alluded to. Soviet intelligence officers frequently exaggerated the success of their operations in the post-Soviet era (ditto the CIA), something that again is easy to do when actual proof is not forthcoming.

It is patently clear that the Soviets and the USA both funneled a lot of money into non-governmental "movements" that either directly or indirectly benefited the position of their governments. Did that "matter"? If so, how much? Are we to imagine a "null hypothesis" scenario in which no foreign money was used in this way? On what evidentiary basis?

If the question is, were the people involved in the peace movement, the anti-nuclear movement, etc., in the Western states (both the US and Europe) earnest in their convictions — the answer is plainly "yes" on the whole. If the question is, did these evolve into mass movements motivated by real feelings of tension and danger — the answer is plainly "yes" on the whole. If the question is, were these movements occasionally boosted by outside funding — the answer is probably "yes" but again it is not always easy to track this. We know about CIA's efforts more than the KGB's because of things like the Freedom of Information Act, and in most cases it is not at all clear that CIA sponsorship of "movements" or cultural activities made much of an impact on any kind of tangible policy outcomes (something that even the CIA eventually concluded).

To add one more wrinkle: it became a common way for the political right to ignore, discount, or attack anti-nuclear movements in the 1980s in particular by claiming they were KGB-sponsored. It was alleged many times that things like "nuclear winter" was a KGB hoax (despite it being pretty clear that, whatever its merits or flaws as a theory, it was not that). So you'll see lots of people who happily take up that line today with the idea that you can discount mass movements because they may or may not have had intelligence funding sources (even funding sources that, in most cases, the heads of the movement may not have realized were ultimately intelligence services, because such services rarely funded "overtly," and instead used shell organizations). It also applies to the Civil Rights movement and anti-McCarthyism as well, for whatever that is worth — the Soviets realized that it was possible to contort American politics by emphasizing these (quite reasonable in retrospective) positions that happened to be quite controversial at the time.

Then as in now, this is a bad way to think about whether someone has a legitimate point or not. The claims about Russian influence in the current elections are that they amplified anti-Clinton and pro-Trump arguments and that this might have had a very small effect (but just enough) to contribute to the already-tight race. Right or wrong, that is not the same kind of thing we are talking about when we ask whether or not various mass movements were directly or indirectly pushing a "Kremlin line" in the 1970s-1980s.

I would want to see a lot more proof because I accepted a former-KGB statement as to the effectiveness of something they did (ditto former-CIA) as a fair statement. But even if one did accept such things, the import of them is probably more complex than a simplistic reading would get you.

(Audra Wolfe's forthcoming book, Freedom's Laboratory, from Johns Hopkins University Press, contains a lot abut CIA funding of a wide variety of activities, and the difficulty in ferreting out the utility of it. Lawrence Badash's A Nuclear Winter's Tale has a lot about the nuclear winter episode.)