Unquestionably. Alongside their talent for HUMINT operations, psychological/propaganda operations ("active measures" in Soviet parlance) were one of the key competencies of the Soviet intelligence services. There's plenty of evidence from defectors that the KGB and GRU funded, supported and manipulated trades unions, counterculture groups and left-wing political organisations around the world; using them to channel disinformation and propaganda in an attempt to create dissent. Not to say that they were always successful, but they were certainly prolific.

From the autobiography of Stanislav Lunev, a GRU (Soviet military intelligence) officer who defected to the US:

Our instructors also told us about how the GRU influenced the American public. The GRU and the KGB helped to fund just about every antiwar movement and organization in America and abroad. Funding was provided via undercover operatives or front organizations. These would fund another group that in turn would fund student organizations. The GRU also helped Vietnam fund its propaganda campaign as a whole.

What will be a great surprise to the American people is that the GRU and KGB had a larger budget for antiwar propaganda in the United States that it did for economic and military support of the Vietnamese. The antiwar propaganda cost the GRU more than $1 billion, but as history shows, it was a hugely successful campaign and well worth the cost. (p78)

The KGB also attempted to infiltrate and co-opt the civil rights movement — they were initially confident that they could covertly manipulate Martin Luther King, but after failing to do so, instead attempted to discredit him. From The Sword and the Shield by Christopher Andrew and KGB defector Vasili Mitrokhin

King was probably the only prominent American to be the target of active measures by both the FBI and the KGB. By the mid-1960s the claims by the CPUSA leadership that secret Party members within King’s entourage would be able to “guide” his policies had proved to be hollow. To the Centre’s dismay, King repeatedly linked the aims of the civil rights movement not to the alleged worldwide struggle against American imperialism but to the fulfillment of the American dream and “the magnificent words of the Constitution and the Declaration of Independence.”

[...]

Having given up hope of influencing King, the Centre aimed instead at replacing him with a more radical and malleable leader. In August 1967 the Centre approved an operational plan by the deputy head of Service A, Yuri Modin, former controller of the Magnificent Five, to discredit King and his chief lieutenants by placing articles in the African press, which could then be reprinted in American newspapers, portraying King as an “Uncle Tom” who was secretly receiving government subsidies to tame the civil rights movement and prevent it threatening the Johnson administration. While leading freedom marches under the admiring glare of worldwide television, King was allegedly in close touch with the President.