The United States fought a war against the Japanese in the 40's, the Koreans in the 50's, and the Vietnamese in the 60's/70's. Was there any amount of extreme racism or leftover hateful sentiment against Asians in general in the 80's and 90's?
Yes, but.
First off, Asian immigrants had long faced resistance throughout US history. The mid-1800s was the beginning of anti-Chinese sentiment on the mainland and (while not yet a state) anti-Japanese sentiment in Hawai'i. The 1880s saw the beginning of decades of anti-Chinese law (the Chinese Exclusion Acts) which lasted until the 1943 (but was still heavily limited until the 1960s). While Japanese immigrants were targeted somewhat later (1924), they too were restricted in the mainland United States.
Additionally, there has also long been an overt identification of certain groups being "unwilling" or "unable" to assimilate to the American way of life (whatever that meant for a given person in a given time in history). You see this same thing with Muslim immigrants being targeted as unassimilable. The same has been said about East Asian immigrants and many other groups. They were criminals, corrupt, not the kind of people we want coming in, etc, that yes some were victims but many were just opportunists. Remember that because I'll bring it up again.
Finally, it's important to understand that, aside from obvious racism for racism's sake, much of this sort of large-scale anti-whomever sentiment, at least as it has been manifested in the US, has been closely tied to economic downtown, competition for jobs, and other tensions that develop when it comes to making ends meet.
What follows is mainly about Vietnamese immigrants in the United States. If you want to know more about historical anti-Asian sentiment I'll be happy to talk about the issues faces by the Chinese and Japanese communities starting in the mid-19th century but for your specific question I think the case of Vietnamese refugees/immigrants is more applicable.
Let's start in the 1970s. During the aftermath of the Vietnam War there were a number of refugees who had been accepted for admission to the US by the US Government. Among the general American public, this was not immediately accepted without complaint, and most of those surveyed did not support taking in Vietnamese refugees according to a 1975 Gallup poll. Before the end of the war, it was more common to hear Americans saying the refugees should be supported. However support for the plight of refugees during the war looked very different from support once the war was over.
Many were still brought to the US, though less than had initially been planned for. Immediately following the end of the war, around 65,000 South Vietnamese were evacuated with help of the US (though with a considerable amount of corruption and extortion going on in the process). They were mostly evacuated to Thailand as a temporary stopover (a frequent temporary stopover for Southeast Asian refugees). In the end around 75,000 got to the US in that year
I mentioned the kind of claims you're probably seeing today on the news about Syrian immigrants and how they're a bad element, wolf in sheep's clothing. In fact not only were this same things being said about Vietnamese refugees at the time, but also many in government expressed these opinions as well. If the US accepted South Vietnamese refugees, the concern was that it would be criminals and not fine upstanding future Americans that would be coming in.
Then there are the economic concerns. Early, even as early as 1975 there were strong concerns that Vietnamese refugees would hurt an already suffering the domestic job market. Despite mixed responses, from 1975 to 1978 when a much larger wave of refugees came, people talked about the risks and fears, but the refugees themselves faces a less fierce sort of opposition upon their arrival in the US, at least among the general public. There was opposition, but it only really picked up after the refugees became much more visible to the average working-class American.
The reasons for this opposition are nothing new. One of the biggest factors was the eternal myth of the un-assimiilable immigrant, which I mentioned above, the idea that Vietnamese coming into America would never really be fully American. This exactly echoes what Americans were saying 100 years before at the beginning of the period of Chinese Exclusion which officially ran from 1881-1943 but had waves much further back and forward. America has historically been upheld as a great melting pot, despite a well documented history of immigration policy and public opinion presenting a contrary position.
There was another significant factor in these attitudes as well, which again echoes the history of 100 years prior. In the 1970s the economy wasn’t in the greatest condition. People felt pressures of unemployment, and the growth of Japanese corporations as competition contributed to an already growing anti-Asian sentiment among nativists.
The state kept both of these issues in mind when the first group of immigrants came, and took steps to lessen the public outcry, but it didn’t really work, and there were a number of incidents over the coming years.
There were somewhat frequent minor scuffles (I’m not sure what to call them. Episodes of violence less that murder or hospitalisation). People were beaten up or their properly damages. Shops owned by Vietnamese were targeted, and often treated unfairly when not otherwise being harassed. It was especially problematic in Texas. Again, as in California a century earlier, the local white population felt growing resentment for the hard working Asian immigrants who were used to poorer conditions and who, due to significant differences in the structuring of social/kinship networks, were able to employ a much larger number of people, free of cost (being that they were family). People accused them of being communist because the "free labour" was seen as an indicator, never mind that many of these refugees were refugees in the first place because of their militant opposition to communism.
Shrimping was the main occupation of Vietnamese in Texas at this time, and in 1984 tensions between Vietnamese immigrants and the white population lead to the burning of multiple Vietnamese-own shrimping boats. This lead to fighting and one native-born American was killed.
However this was not the end of it. Soon the KKK was involved and began targeting Vietnamese families. Eventually this spread to places like New Orleans where a gunfight broke out, again among Vietnamese and local shrimpers.
The anti-Vietnamese sentiment had no trouble in being re-positioned as industry protectionism.
There were plenty of other instances of violence as well as well. In 1989 in central California a man shot up a playground with an AK-47 killing five children and injuring 29 more. Most of the victims Southeast Asian immigrants.
I mentioned that this would mainly focus on Vietnamese immigrants, but in the early 1990s you started to see an increase in Anti-Korean sentiment as well. Not directly tied to the Korean War, but instead more due to the other factors mentioned above (economic competition, general racism). Then in places like LA there was anti-Korean sentiment, leading to the 1992 LA riots, and the Vietnamese were also lumped together with the Koreans and were thus made targets, as they had been with the Japanese in the years before. Of course by the 1990s, the Chinese were seen as successful by white Americans — the so-called "model minority" — while Vietnamese were very much seen as not yet there, and for Koreans, there was a less clear position, but then more often than not Koreans were lumped in with the Vietnamese (and vice versa, the Vietnamese were lumped in with the Koreans).
By the mid-1990s things had settled down more, but still, 20 years of conflict as refugees from decades of conflict back home is nothing to shake a stick at.
I want to add that not everyone in these communities faced the same degrees of violence, suspicion or negativity. There are a number of memoirs that speak more positively. The violence described above isn’t the only story. It’s just the one you asked about.
See also:
(Also for the record I've recycled a good portion of this answer from past answers I've given on related questions)