This is a fantastic and complicated question!
There is a worthwhile backdrop to the 1960s, which is that in the 1920s homosexuality was, seemingly, legal. Western commentators of the 1920s often took note that sodomy, which was illegal in the Russian Empire, was rather conspicuously left unaddressed by the Soviet constitutions. The implications of this are complicated, but I read it as more an effort to reject the Empire wholesale rather than particular support for homosexuality. Indeed it does not seem that decriminalization was a marked change; the historian Dan Healey notes an instance in 1922 (the same year of the constitution) where a group of homosexual men were charged due to the perceived 'social danger' of permitting homosexual behavior. Bekhterev viewed homosexuality as the result of poor environment (and made note to the homosexual behavior of prison inmates in support of this belief) and resultantly argued for psychological cures; and, in an archetypal Soviet characterization, homosexuality became allied with bourgeois values.
The Silver Age poet Mikhail Kuzmin, for example, was publicly homosexual, but very much a bourgeois, decadent figure in the Petersburg avant-garde, allied variously with symbolism and Acemism, and came from an aristocratic backround. (In the 1920s he continued to publish poetry, but is probably more well known for his modern translations of Shakespeare).
Punctuating the Cultural Revolution is the recriminalization of homosexuality (sodomy) in 1933, which is characterized by the historian Richard Stites as a reactionary, grassroots rejection of bourgeois - by now anti-Soviet values - as a "Sexual Thermidor". In the 1930s homosexuality remained strongly linked to bourgeois values, and as ex-aristocrats were characterized as 'former people' homosexuality becomes dehumanized and advocates arrested or jailed.
After the death of Stalin there was a reconsideration of, well, lots of things, including criminal codes. Homosexuality remained criminalized in 1960, but there was no longer a minimum punishment for imprisoned (but a maximum of five years). The criminologist Mikhail Khlyntsov characterized sodomy as a crime in 1965 as causing "extreme danger [by] its encroachment on the moral foundations of society." Throughout the 1960s there were considerable debates on how to frame understandings of homosexuality: as a criminal concern or a medical (redemptive) one.
So what would life be like for a gay man in the Soviet Union of the 1960s? You are likely, if caught, to be tried and jailed; on occasion it does seem that sexual acts outside the purview of sodomy (that is, penetrative sex) were viewed as 'depraved' but legal, while other cases for similar instances seem to have been charged under a more expansive view of 'sodomy.' Poet Gennady Trifonov was jailed for his gay poetry of the 1970s, and remained under suspicion throughout the 1980s. There were active, underground gay communities, most probably related to the prison system due to the permissibility of homosexuality (oftentimes, but not always, homosexual rape). It's not until the 1980s that agitation for gay rights and gay groups swells. Homosexuality is now legal in Russia and has been since 1993, but is still culturally unpopular and recent laws have focused on censorship of LGBTQ activism.
There is a forthcoming book by Rustam Alexander entitled Regulating homosexuality in Soviet Russia, 1956–91 that may be of interest as well.
References:
Alexander, Rustam. "Homosexuality in the USSR (1956-82)." PhD diss., 2018.
Healey, Dan. Homosexual desire in revolutionary Russia: The regulation of sexual and gender dissent. University of Chicago Press, 2001.