Tailin Lyngdoh, who is from Meghalaya state, was wearing the traditional Khasi tribal dress. She had gone to the Delhi Golf Club with her employer.
The staff asked her to leave because they said she looked "like a maid".
The club said the "incident could have been handled better" and are taking disciplinary action against staff.
"We have sought an explanation from the staff and disciplinary action is in process," the club said in a statement.
"An apology has been made to the member who had brought the guest. This has been unconditionally accepted," it added.
Reports said Ms Lyngdoh, who works as a governess for Dr Nivedita Barthakur Sondhi, visited the club on Sunday after they were invited by a member.
"Around 10-15 minutes after we arrived, the manager, Ajit Pal, accompanied by a woman, asked Ms Lyngdoh to leave the table and the room," Dr Sondhi was quoted by the Times of India as saying.
"When I asked them the reason, they said she looks like a maid. I asked them how they concluded that? They said she looks different, dresses like a servant and looks like a Nepalese. That was so humiliating. I wasn't ready to accept such discrimination," she added.
Dr Sondhi said she protested "as they were insulting a traditional dress of an Indian citizen. I also argued that her profession is immaterial as she has been invited as a guest by a member".
The apology came after many took to social media to criticise the club for being insensitive.
An elite Delhi club has apologised after its staff told a woman who turned up dressed in a traditional north-eastern Indian outfit to leave.