What if LBJ had decided against the Vietnam War draft? What influence would that have had on US public's view of the Vietnam War?
What do you mean? Only sending voluntary enlistees to Vietnam? Draftees were intermingled throughout the Army, so that's not practical, unless some special commands were established.
Only sending men to Vietnam who volunteer or consent to go? That's even trickier.
Both policies reduce the strain and loss to the U.S. They also mean, IMHO, a higher quality of men in Vietnam, and much less problems with bad acts by U.S. troops there - which surely changes the popular view of the war.
If the second policy - it will be extremely difficult for the U.S. to deploy any large number of ground troops in Vietnam. There will be no question of U.S. forces winning the battles against the NVA/VC; instead it will be necessary to train up ARVN forces. Green Berets doing COIN will be more important as well.
If North Vietnam pushes with so many NVA that they threaten to win; more U.S. airpower, and more airpower against North Vietnam. Also, there may be opportunities for the U.S. to deploy ground troops in places like Khe Sanh and the Highlands and other places to provide secure bases and strongpoints, but not COIN.
The U.S. did train up ARVN to be much more effective in about two years. The problem was that MAC-V didn't really start on that until 1968, when it was already too late for U.S. morale.
The crunch would be in 1964-1967, with the NVA trying to hit hard, few U.S. troops to meet them, and ARVN not ready. A Tet-level offensive could come close to winning.