Demographics obviously do play a role in music taste, and there is obviously psychological research demonstrating this. Other demographics in the US that play a role beyond ethnicity/class/conservativeness/etc also are geographical - there's differences in the South, where country music is much more common, for example, and more integrated into society's social structures (e.g., plenty of country stars of the 1960s supported George Wallace, and put on a benefit show for him at one point, and a big part of the history of Nashville was its attempts to move from a perception of country music as 'hillbilly music' to something more 'countrypolitan' and respectable).

Nonetheless, there was likely a variety of diversity in the tastes of more conservative people in the 1980s, still, because such demographics aren’t the only thing that influences musical taste - of course, peoples' personalities do too. A more extroverted, outgoing person is probably more likely to like dance music than melancholy singer-songwriter stuff, for obvious reasons (though not I'm using words like 'likely' here - these are trends, and not fixed hard rules, because people are complex and contain multitudes etc).

Broadly speaking, in the 1980s, the music considered countercultural (for white Americans anyway) was the broad spectrum of music from hardcore (e.g., Black Flag) to indie (e.g., Sonic Youth or REM) to new wave (e.g., Blondie) to post-punk (e.g., Public Image Limited). By countercultural I mean that this music very often opposed mainstream culture from what in the 1980s was considered a left-wing perspective; these people considered mainstream pop music to be hopelessly conservative. So yes, it would likely be relatively rare for 'punk' music to be their music of choice - in the 1980s, this would have been music they would have needed to seek out, to some extent, as it was much less likely to be represented in mainstream media.

Instead, what they're likely to have listened to is music that was considered relatively mainstream and safe by those in the counterculture. The stuff on the radio that people ignore in shops, or whistle under their breath, but things that nobody thinks that hard about. By the 1980s, for someone young, that was very likely 'rock-type' music rather than, say, Frank Sinatra (though their parents might still have listened to music like that, and that might have influenced them).

One satirical literary look at a conservative American milieu is of course Bret Easton Ellis's American Psycho, which the later film of the same name starring Christian Bale was based on. The protagonist in the novel famously discusses his musical tastes in amongst a variety of narcissistic behaviour and descriptions of horrific events they're inflicting on others. Ellis, who does come from a 1980s countercultural background (his first novel, Less Than Zero, was titled after a song by Elvis Costello, an artist then strongly associated with punk and new wave), had his protagonist rhapsodise about artists like Huey Lewis and the News, Whitney Houston, and 1980s Genesis (but not the progressive music Genesis released in the 1970s - instead more the more straightforward pop music version of the group).

From another perspective, the (prominently young Republican) character Alex Keaton in the 1980s TV show Family Ties, in a 1987 episode called 'My Name Is Alex', is seen to lip sync to songs by the 1960s groups Steppenwolf and The Doors, despite usually claiming that their favourite music was swing and big band. Again, Keaton is a character from a TV comedy, but it does demonstrate what people at the time thought were likely to listen to; Steppenwolf and The Doors were likely closer to the musical tastes of Keaton's hippie parents, and the TV show was illustrating that his parents had affected who he was at heart, in that episode. Those who came from a upper middle class professional background were more likely than others to listen to, say, classical and jazz music (as Keaton does when he claims to like big band music), but this is only a tendency rather than a hard fixed rule.

But sometimes the more musically engaged people forget that most people are not as strongly musically engaged as they are, and have much less strongly held and much less carefully articulated and thought-out preferences. Very often when conservative politicians talk about their musical taste, they're surprisingly likely to mention favourite bands being bands with quite opposite ideology. The Australian conservative politician John Howard, elected Prime Minister in the 1990s, famously claimed his favourite artist was Bob Dylan, but that he didn't really listen to the words. Paul Ryan, the Republican Vice-Presidential candidate in 2012, famously claimed he was a fan of 1990s alternative rap-metal group Rage Against The Machine, a group with some fairly strongly left-wing views. Mostly, chances are that the young conservatives you mention were likely listening to the same enormously popular pop acts of the 1980s as everyone else - Bruce Springsteen, Madonna, Michael Jackson, U2, Whitney Houston, etc.