When will the next transit of Earth be visible from Mars? Was the last one really on May 11, 1984?
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.