The threat of nuclear conflict was very real during the Cold War. I remember my teacher asking the class (in 1984) who thought that a nuclear war was "likely in the next 10 years" and virtually every hand going up. Some teenage angst there perhaps, but it was as high in the collective conscious then as climate change is now.
People tend to forget how close we came to all out nuclear war on at least two occasions (Cuban Missile Crisis and Able Archer 83) and that there were military leaders in the west in the 50s and 60s who actually wanted to initiate a first strike.
1987 is towards the end of the war though. Arguably, in 1987 the USSR was already in the process of collapse. The last attempt of saving it - Perestroika - was initiated in the same year, due to obvious and catastrophic systemic economic problems, and failed miserably very soon. By 1989 the international power of USSR was largely gone, and in another 2 years, USSR was no more. While in 1984 the concern may be very real, in 1987 more people should have seen where it's going. Not many did so.
I was lucky enough to grow up a few miles from an Air Force Base. When we did the "duck and cover" training thing, our science teacher told us it wouldn't matter because we'd turn to ash almost instantly anyway.
Recently there was mention of a poll of evangelical Christians and some large percentage thought the rapture would happen in their lifetime.
Honestly, that's pretty much literally the same psychology at work as those who believed in impending nuclear holocaust then and imminent climate doom now.
There's been a bunch of mini-dooms along the way as well, everything from y2k causing planes to drop from the sky to terrorists attacking cities more frequently.
I think it's in peoples nature to mentally take trend lines straight up, or even exponentially up. So, if there were x nuclear warheads in 1987, then they assumed it would be 100 * x in 2012.
The same pattern is at work with the climate now, as you've pointed out. Those who speak of the largest catastrophes get the most airtime, and it's a self-reinforcing cycle. Any room full of teenagers will inevitably predict the world will be uninhabitable by their 40th birthday, when in reality it will probably be much the same.
