It was possible, but it was very unusual.

First a quick word on Soviet travel: officially, there were two major types of vacationing for Soviet citizens: otdykh, which roughly translates to "rest", and turizm.

The former type of vacation was to a sanitarium, spa, rest home or some other type of facility, usually in a particular area known for natural beauty, clean air and the like. The idea of Soviet citizens staying there was that workers would be able to recuperate from their work, and that this recuperation would in turn make them more productive on their return to work.

Turizm is, of course, tourism. The idea here being that Soviet citizens would travel to a particular place of cultural or historic note for edification and education, as well as camping and strenuous physical activity. One area in particular that tended to attract a lot of organized travel domestically in the USSR was the Baltic, especially Riga, Tallinn and the Latvian and Estonian coasts, were something of a domestic "Europe" Soviets could travel to and admire for their history, and architecture, although by the 1980s more tours were taken abroad than domestically. The Black Sea, especially Crimea, Sochi, Abkhazia and Georgia, were more destinations for sun and sand and for otdykh at spas (it's worth noting that Abkhazia in particular has a subtropical climate). Other forms of group tourism that were undertaken in the Soviet period were "scientific tourism" (ie, scientists and specialists traveling for specific purposes outside of international congresses), and "Youth Tourism" (especially for such mass international events as the World Festival of Youth and Students, which was held in Moscow in 1957 but in other locations before and after) and international exchanges.

It's worth noting that vacationing in both contexts was generally set up as group activities: namely, if you were staying in a sanitorium, or on a tour, you did it as part of a group. Soviet did of course get their vacationing in in more individualistic ways, but this was more "off the books", and probably would involve spending the weekend or holidays at a dacha (country house), which may or may not have been privately owned by a particular citizen or his/her family (the rules differed by time and place). Of course even spending time at a dacha had a productive aim as well, as a large amount of gardening was usually done there.

Just to give a sense of the figures: in 1980, out of a population of approximately 262 million, an estimated 188 million people took a holiday away from home. Of the 40 million Soviets taking vacations at organized vacation destinations, some 17.5 million were staying at spas or rest homes, and some 22.5 staying at tourist bases.

Some 4.3 million Soviets traveled abroad in 1979. It's worth noting that most of this travel was to "friendly" (especially socialist) countries, but all international travel involved a process of being approved for an exit visa to authorize travel abroad. This involved applying for travel with one's Trade Union or Komsomol (Communist Youth) branch. Then a provincial party committee, usually with commissions of party, trade union, and KGB representatives, would review the application (with special attention on those traveling to capitalist countries). Once a Soviet citizen was accepted for travel abroad, a list of people assembled for a group, complete with character assessments, completed questionnaires, medical forms and a payment to Intourist's State Bank account, were forwarded to Intourist for processing. Those who were approved for travel often (if not always) had some sort of party connections (academics, factory managers, functionaries, etc.). State-provided vouchers would cover travel expenses, and an individual approved for travel was just that - an individual approved to join a group. You didn't take friends or families along.

The top 10 foreign destinations for Soviet travelers in 1980 were, in descending order, Bulgaria, East Germany, Czechoslovakia, Poland, Hungary, Yugoslavia, Finland, Romania, India and Cuba.

Again, travel to these foreign destinations would be in a group. It appears that the first group of Soviet tourists to Cuba was 20 tourists dispatched there by Intourist (the Soviet travel agency) from Mexico in 1960, but it seems to have been intermittently offered as a route after that (in 1964 and 1969).

So: Soviet citizens did indeed travel to Cuba in the 1970s and 1980s, but it was a relatively rare occurrence, and would have involved travel as part of a group, most likely in some sort of cultural exchange program. Individuals or individual families traveling on their own to sit on the beach would not have been an option - but there were official and unofficial options for that much closer to home anyway.

Sources:

Anne Gorsuch. All This is Your World: Soviet Tourism at Home and Abroad after Stalin

Diane Koenker. Club Red: Vacation Travel and the Soviet Dream

Igor Orlov. "The Soviet Union Outgoing Tourism in 1955-1985: Volume, Geography, Organizational Forms". via National Research University Higher School of Economics, here.