Well, Johnson might be reelected if not Tet offensive right ?
It all depends when Vietnam War become stillborn. 1963 ? My favourite alt-history has a POD in November 1963.
Kennedy's dead, but not Diem. As bad as the guy was, he managed to keep South Vietnam afloat without direct US involvement.
So Diem survive and hand SVN long enough to avoid the US intervention. Only to be overthrown later (1965 ? 1967 ?) by a Tet-like riot.
And Johnson is reelected in 1968. Right from 1965, NASA planners start thinking about post-Apollo plans; something to be done as early as the mid-70's. That's too early for a manned Mars landing, and, above all, 1975-1977 were crap years for Mars missions, which works on a 15/17-years cycle (1971, 1986, 2003 were good years).
However that doesn't prevent NASA from planning
flybys. The concept may sound crazy: the ship meet Mars or Venus at very high speed, staying there only for some hours. During this brief time astronauts drop landers, balloons, rovers, sample-return probes.
Here's a neat little concept: a triple planet flyby in the year 1977 or 1981. Earth-Mars-Venus and back !
http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19790072312_1979072312.pdf
Another neat trip was the Eros flyby.
http://beyondapollo.blogspot.com/2009/01/manned-eros-flyby-1966.html
Geostationnary flights
http://beyondapollo.blogspot.com/2009/01/destination-mankind-1972.html
Were also planned flights to Lunar polar orbit (where probes tentatively identified water in 1998)
These flights all used Apollo hardware - kind of first generation, in the 70's.
Second generation hardware was even more ambitious.
It consisted of
- the space shuttle, in a fully reusable form
- a 10 > 50 > 100 man earth orbital space station
- nuclear engines, the NERVA series.
Nuclear-powered shuttles would replace Apollo for lunar flights, and allow manned Mars landings, in 1981 (earliest possible date, maximum funding) or 1986 (relaxed funding and planning).
For the Mars landing itself a new vehicle would be developped, the MEM, Mars Excursion Module.
What would have been feasible if no Viet-Nam war ? I'm not sure the "second generation" would have been developped. NERVA, shuttle, 100-man space base, MEM and manned Mars landing were quite costly and ambitious.
Extended "first generation" hardware looks more reasonable. More Apollos, extended lunar missions with more Saturn Vs, maybe the shuttle flying to improved Skylabs.
In conclusion: without viet-nam, maybe no manned Mars landing, but certainly more lunar missions...