With regard to the Left versus the Right and their support of 'the military', I think you have to first state your framework.
Era:
In terms of era; I am a post Vietnam-era veteran, having enlisted in 1979. The military was not looked upon fondly by the Left in my experience at that time. I'll leave out the histrionics and the 'baby killer' remarks or accusations of being spat upon by hippies; suffice to say I did not experience that myself, but I certainly did experience the attitudes of the times. We were less than respected; little more than low-brow louts. The military was where one went if one had no hopes of college or a minor criminal background and did not wish to 'make something' of oneself. Others may disagree, but that was my experience. The Right by comparison, was broadly supportive of the military, also in my experience.
Prior to my era, the Vietnam era veterans had it worse. It was my understanding from the time I served with them (many were still on active duty when I was) that they experienced so much discrimination and anger directed towards them by the anti-war groups (and yes, that would be the Left) that many of them had a permanent dislike for all things Left-wing. By the same token, many of them had also experienced the pain of being rejected by their fellow veterans of WWII and Korea as being 'losers' in war, and 'dope-smokers' and 'hippies' themselves. They were ostracized, even excluded, from the Veterans of Foreign Wars, and when they were finally allowed to join, it was made clear that they were not wanted. This, plus the fact that many Vietnam-era veterans were not in favor of the war in the first place, led them to found their own group, the Vietnam Veterans of America, and to dislike the Right, which the WWII and Korean era vets seemed to mostly be. Please note that today, many of those animosities are gone, thankfully. Vietnam veterans are rightfully seen as the heroes they are.
The WWII and Korean era veterans were, to the best of my knowledge, often members of the Right wing themselves. They had been raised to hate and fight communists and communism. Yes, we fought alongside the USSR in WWII, but not happily, and the way Berlin and Eastern Europe were divided after WWII was proof of that. Many today do not even realize that in the Korean war, we fought Chinese regular troops face-to-face. It was no proxy war, the UN was fighting Red China (as it was known to us at that time). Of course, few civilians had a negative opinion of the military following WWII or Korea. WWII veterans were seen from the beginning as the 'Greatest Generation' of Americans, and Korean veterans were largely forgotten, but certainly not despised or hated.
War:
One must also consider the attitudes of the Left and the Right with regard not just to the troops, but also to the conflict. Basically, the Left was pro-war in WWII, ambivalent during Korea, anti-war in Vietnam, and anti-war (but claimed to be pro-troops) in recent conflicts. There are some exceptions. The Left was pro-war with Afghanistan following 9/11, but anti-war with Iraq during the same time period. The Right has been steadfastly pro-war (as far as I know) in all our military conflicts since WWII (although both the Left and the Right were anti-war prior to WWII).
Anti-Troops versus Anti-Military
I think one must also consider that there is a more nuanced approach that tends to be independent of political leanings, and that is how one thinks of the military versus how one thinks of the people who serve in the military. I have met enough people to know that many do not like the military in general, but they have no animosity towards those who serve in it. It's rather like being anti-union, but not disliking people who drive trucks or build cars. It can be confusing, however, when one is not sure what one is hearing. Anti-military rhetoric can sound a lot like anti-troop rhetoric, unless the speaker makes it clear that they mean one and not the other; and some people have a natural suspicion that others can actually separate the two. For example, in the most recent conflict in Iraq, many on the Left (Moveon, etc) were demonstrably anti-war and anti-military, but they claimed to 'love the troops'. This was sometimes not true (in my opinion) and sometimes true but not believed by others. There was so much dislike and distrust between the Left and the Right in those cases, that few believed the claims of the others, and tended to lump a dislike of the military or the war into a dislike of service personnel and veterans. The fact that some veterans were also members of the Left and anti-war only heightened the distrust and dislike. One also has to remember that although it is not commonly brought up, many members of the Right are technically anti-military also, as the more Libertarian-leaning Right see two problems with the military. One is the prohibition in the Constitution against a 'standing army' initially, and two is the warning against a strong military-industrial complex that Eisenhower warned us of which undermines Democracy.
In summary, I do not think the Left is generally anti-veteran or anti-military or anti-troops. However, they do tend (in the USA) to be anti-war, which sometimes doesn't come across as nuanced as it could, and there are some who truly do show animosity towards servicemen and servicewomen (and they tend to be seen as representative by some on the Right), and they often seem anti-military by contrast with the Right, which is typically not anti-war, anti-military, or anti-troops.