I would have to say that biggest lingering influence would have to be the effect the Vietnam War had on the lives of the men that served in the war. Many of them have been scarred either physically or mentally for life as a result of the Vietnam War.
Certainly, after the government conspiracy about the Bay of Tonkin incident which was the excuse for going full throttle into Vietnam, there is much skepticism on the part of older Americans, especially, regarding U.S. involvement in conflicts in the Middle East. [Former Minnesota governor and Navy Seal Vietnam veteran Jesse Ventura has written a book on this and other conspiracies: American Conspiracies]
In addition, it is worthy of note to mention that there is a higher percentage of Vietnam veterans in prison than any other veterans of other wars/conflicts.
Keep in mind that roughly half of the US population was born after the war, and many if not most Americans are poorly educated on this country's history. So there is a serious limit on the influence you refer to.
There is a direct influence in that we have millions of surviving veterans of that war, and tens of thousands of surviving wounded, some which require medical care to this day. These veterans have raised families, and passed on their lessons and experience to their kids.
That being said, the influence of the national nightmare that was the Vietnam War was not enough to limit American support for the invasion of Iraq, at least initially.
I would say that there are three major lingering influences of the Vietnam War today:
- First, I would say that the Vietnam War has helped make Americans trust their government less than they used to. During the war, the government seemed to be lying to the people and that helped ruin the trust people used to have in the government.
- Second, it has helped make Americans more skeptical of going out with the military to try to help other countries. The Vietnam War showed that it was hard to use the military to make people like us. This effect may have worn off to some extent, seeing as how we are somewhat in the same situation in Afghanistan.
- Finally, I would say that it makes Americans more zealous about honoring the military. The reception that Vietnam vets got was shameful and I think that we are trying to make up for it with how much we talk about honoring "the troops" today.