When will the next transit of Earth be visible from Mars? Was the last one really on May 11, 1984? @PM2Ring's comment mentions the story Transit of Earth written by the famous British science fiction writer, science writer and futurist,3 inventor, undersea explorer, television series host, Fellow of the Royal Astronomical Society and Commander of the Order of the British Empire and celebrated polymath Arthur C. Clarke in which the character's task is to record a transit of the Earth across the Sun's disk as seen from Mars.
According to the story, this happens on May 11, 1984.
The story also mentions that the previous occurrence would have been in 1905, and the following one would be 100 years following this one, presumably in 2084.

The radio has just printed a message from Earth, reminding me the transit starts in two hours, as if I’m likely to forget, when four men have already died so that I can be the first human being to see it, and the only one, for exactly a hundred years.
It isn’t often that Sun, Earth and Mars line up neatly like this. The last time was in 1905, when poor old Lowell was still writing his beautiful nonsense about the canals and the great dying civilization that built them.

Question: Were there in fact transits of the Earth as seen from Mars on these dates (in 1905, on 11-May-1984, and in 2084)? If they do occur at roughly 100 year intervals as suggested, there may in fact be people on Mars to see the next one presuming that Sir Clarke did his maths right (did he ever not?). On what day would that be?

You can listen to Sir Clarke read the story himself in the YouTube video 'Transit Of Earth' by Arthur C. Clarke read by himself. You can also listen here: http://recordbrother.typepad.com/imagesilike/files/transit_of_earth.mp3 and it seems download HMIcomposite_2Kx2K.mov from NASA here (though I haven't verified it yet): https://svs.gsfc.nasa.gov/vis/a000000/a003900/a003941/
Yes. The Clarke story is quite correct. Earth (and Moon) would transit the sun from the viewpoint of Mars on 8 May 1905, 11 May 1984 and again on 10 Nov 2084.

full list over a period of about 1000 years, from Wikipedia Transit of Earth from Mars 

November 10, 1595   May 5, 1621     May 8, 1700     November  9, 1800   
November 12, 1879   May 8, 1905     May 11, 1984    November 10, 2084   
November 15, 2163   May 10, 2189    May 13, 2268    November 13, 2368   May 10, 2394
November 17, 2447   May 13, 2473    May 16, 2552    November 15, 2652   May 13, 2678


They follow a pattern of occurring after 26, 79 then 100 years.