This is a perennial issue and there are scores of threads about this. I’ve worked with, and known, many Barnard and Columbia students and any “rivalry” is insignificant. Indeed, other than Freshman seminars, it is rare to find Barnard courses that are all female. And we just hired a Barnard grad who majored in Mandarin and she said that virtually all her classes had been at Columbia. When she liked about Barnard was its smaller administration and the Athena Women’s Leadership programs.
I guess my plea is to look at past threads rather than reverse a tiresome subject. Suffice to say, Barnard is very different from Smith, Mt Holyoke, Sweet Briar etc as Columbia is literally across the street.