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How would a plasma engine work?

Badly. The problem isn't with the plasma engines themselves, but with the power plants. In order to achieve the 39 days to Mars claimed by NASA, a VASIMR engine would need to be coupled to a nuclear reactor system with a power of 200,000 kilowatts and a power-to-mass ratio of 1,000 watts per kilogram. The best reactor achieved to date, the Soviet Topaz, had a power of 10 kilowatts and a power-to-mass ratio of 10 watts per kilogram.

To make matters worse the anti-nukers have put the screws to reactor research. So even if the power-to-mass ratios specified by NASA were possible, and a significant number of scientist and engineers doubt that, we won't be able to build and fly the experimental reactors necessary to find out.

Besides the VASIMR ship gets real big real fast. Can't launch it in one lift. So now you not only have to build a Battlestar, you need to build and orbiting shipyard too. This in an era where NASA can't get the budget to fly, well anyone.

The example I use is the 747 vs the Conestoga wagon. If you are going from St. Louis to California the 747 is faster, cheaper per passenger, safer, more reliable, moves more cargo and vastly more comfortable, even in coach. But the Conestoga departs in 1849, the 747 doesn't fly until 1969. And that 120 year head start is going to make all the difference.

You are better off going with smaller, lower tech, lower budget missions. Proper design can use the Earth departure stage as a counterweight to spin up the habitat module for gravity. And you can use to stores, water and fuel as a storm shelter against cosmic rays and solar flares. Yes you are going to Mars in a tuna can. And you spend more time in the can than the battlestar. But the tuna can is possible immediately within technical and budgetary restrictions. VASMIR might be used to colonize Mars some time in the late 21st or early 22nd century. But if you want to explore the place before then big dumb boosters and methane rockets are a much more workable solution.