By the time I came of age, there was no draft, but I still had to register for Selective Service (which was brought back by Reagan). I attempted to enlist in the US Navy when I was 19, but I failed the physical. So, at least I can say with a clear conscience that I tried to serve but was turned down.
I see nothing wrong with being a conscientious objector, although that wouldn't have necessarily exempted one from military service. One can serve in non-combat roles.
I had somewhat mixed feelings about war when I was younger. I was born later than the average hippie or Vietnam Vet. I was influenced by the anti-war movement as I was growing up, and definitely enjoyed the music of the era. But I also had other influences regarding America and war, such as my uncles who fought in WW2 and were pretty much hawkish and right-wing on their view of the world. I've also befriended some Vietnam Vets who also had a rather hawkish view.
But I also knew quite a few lefty types - even more when I got into college. Not very many radicals or revolutionaries by then, as the tumult of the 60s had cooled down considerably. There wasn't really even a very strong anti-war movement either, not by the 80s. It's not as if their ideals changed or their views of the U.S. government had changed, but it just wasn't that important anymore. People just stopped caring. At some point, the military became cool again, and the big thing was "support the troops." Even Jane Fonda apologized.
For similar reasons, it's why it comes up as an issue in political campaigns, as it did with Quayle, Clinton, Trump - or anyone else who didn't serve. The fact that it's brought up at all is an indicator that political pundits obviously believe it will influence a significant portion of the electorate.
It's not so much about war, in and of itself. It's not even about serving in the military. These politicians and others who have ostensibly reversed their previous anti-war stance are, in essence, affirming their belief in the same geopolitical perceptions and national security aspirations which they previously opposed.
In other words, they were right when they younger. I've heard it said that "If you're not a liberal by age 20, you have no heart. If you're not a conservative by age 30, you have no brain."