In my country, the term breadwinner is widely used. However, several years ago, if I remember it correctly, some TV personality said that the term breadwinner is not anymore the usual term to call a person who primarily earns for the family. If this word is not widely accepted nowadays, what word would be rather used?
As Michael said, breadwinner is alive and well in British English. In American English, breadwinner (though still right up there) is falling in use (less than half the British use, and falling; if you use the Ngram, extend to 2013). The word is being replaced by "primary wage earner".
Quite a hubbub was created here recently when a Pew study showed that in 40% of households, women are the primary wage earners.
The Pew Research study mentioned in another answer is titled (ironically enough), "Breadwinner Moms." I would agree another answer that the demise of breadwinner is exaggerated.
While COCA does seem to show that the use of breadwinner has tailed off somewhat in recent years, charting breadwinner vs primary wage earner in Google Ngrams for American English shows that breadwinner is still used much more frequently.