First, the anti-war movement has helped to make our society less respectful of traditional authority. The anti-war movement is not the only factor that has done this, but it helped. The movement hated what the government was doing and held the government in contempt. When this sort of attitude is widespread in a society, it makes it more likely that people will come to have less respect for the government and for authority in general.
Second, the anti-war movement helped to create a deep divide between liberal and conservative Americans. In the 1960s, conservative Americans typically thought that the anti-war movement was made up of traitors. They felt that it was wrong for Americans to question and to try to undermine their government. By contrast, the liberals thought that the conservatives were immoral for blindly following their government in what they saw as an immoral war. This helped create a situation in which liberals and conservatives often despise one another. This is a situation where each side things the other side is morally bad, not that it is simply mistaken. These are two effects that are felt in our society today and which can be traced in part to the anti-war movement of the 1960s.