The Vietnam conflict became increasingly unpopular as time went on. The United States public did not understand why its young men were being to sent to fight in a small country halfway around the world on a land which seemed to pose little or no visible threat to the US.
Along with the civil rights movement and the women's movement, the anti-war movement led to an upsurge in civil disobedience and discontent. The public's unhappiness with US foreign policy, in turn, led the government to implement reforms. For example, though the draft age was eighteen, US citizens at the time were not allowed to vote until they were twenty-one. This caused a great deal of unhappiness on the part of young people who were forced to fight in conflicts over which they had no say. As a result, Congress in 1971 ratified the Twenty-sixth Amendment to the US Constitution, which allowed eighteen year olds to vote.
Also as a response to domestic unrest over the highly unpopular draft, the draft was suspended in 1973 and has not been used since (although it theoretically could be, as young men are still required to register for it). Since Vietnam, however, we have had an all-volunteer army.
Vietnam also impacted domestic policy by cutting into funding for Johnson's Great Society programs meant to bring increased prosperity to middle-class and poor Americans. Wars are very costly, and Viet Nam became a textbook case of throwing money at a problem. This diverted tax dollars from investment in the United States itself. By the time of the Nixon administration, the costs of the war were fueling inflation to a degree high enough that Nixon temporarily imposed domestic wage and price controls. This shook the business community—as it did not like having its pricing interfered with—and was part of the impetus behind the rightward drift of the Republican Party.
Being involved in a military conflict that had shaky public support and vocal opposition has had a lasting impact on domestic policy in the United States.