"Hmm, if getting to Mars through VASIMR would take much less time, this would expose the crew to far less sun and cosmic radiation than a 6 months long trip. A nuclear rocket would reduce exposure, thanks to the time saved."

There is certainly many advantages of getting to Mars quickly. VASIMIR or other types of nuclear propulsion are at least double the ISP of chemical rockets. A nuclear reactor provides a constant source of energy, comparing the energy output of a nuclear reactor operating over a year of time to the total output from chemical energy source, the nuclear reactor will require far less total mass & deliver more energy. If you shorten the time period, to one month, then the nuclear advantage is less dramatic- 1/12th it's "former glory". This is one aspect.
Another aspect is the acceleration required in order to shorten the travel time to Mars. A 9 month trip to Mars require slightly more delta-v then the escape velocity of Earth, 6 month trip significantly more delta-v. Roughly, if it takes 1 km/sec over earth escape for 9 month then it take 3-4 km/sec for 6 month (and continues this exponential growth as you shorten the travel time). The added velocity for 6 months can probably be mostly shed aerobraking using Mars' thin atmosphere. Shorter travel times will require rocket power for deceleration. This IS an advantage for nuclear powered spacecraft in comparison to chemical- if you *need* more total delta-v and can expend this energy over a relatively longer time periods of days to weeks.
My point was rather simple. 4 month travel time to Mars might be doable with nuclear propulsion (it also possible with chemical). 2 month travel isn't doable- at least in the near term. As example, it's possible for a land vehicle to break the sound barrier, but it's not practical to expect that you can have a car go 600 mph for your morning commute- at least in the near term.

"Yes, it's going to be more expensive than a chemically propulsed rocket, but such a rocket will already be extremely expensive anyway. So a fraction more, will that make such a difference? I'm not sure."

I'm not talking about a fractional increase in cost (even though what you *want* to do is lower the total cost, not raise it) it's doubling or tripling of total cost which I'm addressing. A goal of a travel time 4-5 months would be more reasonable. Perhaps the "political problem" with this is, that 4-5 months doesn't exclude chemical rockets.