The Cambridge professor and TV presenter also spoke out against a “snobbish” attitude towards academic subjects.
She told The Times’ Education Commission there had been a “revolution in gender” at universities, with most of her peers as a student middle-class white men and a rise in the number of women at Cambridge during her time there.
But Dame Mary said Cambridge and similar universities “feel very southeastern” and “white”.
“While I’m no defender of grammar schools, when I think back to my own undergraduate career, there were more northern working-class blokes around my faculty than there appear to be,” she said.
“You’ve got a small number of elite educational places and you bring in different groups and they edge out others, there’s a constant competition.”
Dame Mary added: “One of the things I would suggest is that we stop being so damned fixated on Oxford and Cambridge and think about a range of universities in this country. Not worry so much about what is always treated as the peak.”
New figures show a record number of 18-year-olds in the UK – 272,500 – are planning on starting university this year.
It comes after more than 44 per cent of students achieved an A or A* grade in their A-level grades this year in England, the highest figure to date.
Ahead of results day, a charity boss suggested too many students were turning to university and graduating with skills that the marketplace did not want.
Dame Mary said: “This country has been terribly snobbish about what education is for.”
“I spent my life doing Latin and Greek. Actually, I think there’s huge value in that but I’m not going to sit here and say it’s more important for someone to study Latin and Greek rather than practical engineering.
“Most of British culture is still held back by class and privilege, aspirations which rank these subjects into sort of what clever posh kids do and what other people do.”
Earlier this year, the government announced Latin lessons would be offered to thousands of state school pupils in England in a drive to make the language less “elitist”.
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