His father, who belonged to the Basu family of Birnagar in Nadia District of West Bengal,[4] was the Dewan Darbhanga Raj.

Shashisekhar, his elder brother, later wrote that the young Rajshekhar put together a laboratory at home equipped with two cupboards of various chemicals; he would forecast the weather by looking at a barometer that he had hung on the wall, would write prescriptions of cough-mixtures for his family members, and later, would even go to the Temple Medical School to dissect corpses.

After graduating he completed a degree in law as well, but only attended court for three days, after which he quit the legal profession for good, and decided to pursue a career in science.

Around this time, he met Acharya Prafulla Chandra Roy, who had recently started a company – Bengal Chemicals & Pharmaceuticals.

A few years before its publication, in 1935, Calcutta University formed a committee, chaired by him, to formulate a set of guidelines governing the spelling of Bengali words.

Basu's collection of short stories, Anandibai Ityadi Galpa, won a Sahitya Akademi Award for Bengali literature in 1958.

The book was published under his pen name, Parasuram, and contained fifteen satirical stories, touching on themes of love, courtship, families, and politics.

The second edition of Parashuram's Hanumaaner Svapna Ityadi Galpa was the first book to be completely printed in Bengali linotype.

He lived for almost 18 years after his wife's demise and wrote a great deal during this time, but he did not allow his personal tragedies to colour his writing.

Rajshekhar's younger brother, Girindrasekhar Bose (1887–1953), was an early Freudian psychoanalyst of the non-Western world, and also wrote books for children.