“Nobody is expecting a toilet police” but people should follow the rules when guidance is finally issued on single-sex spaces, the chair of the equalities watchdog has said.

Mary-Ann Stephenson said that “generally speaking, we expect people to follow the rules and make sure that there is adequate provision”.

The Equality and Human Rights Commission is waiting for ministers to approve its official guidance on how business and public bodies should respond to a supreme court ruling in April that the legal definition of a woman is based on biological sex.

The EHRC’s guidance was passed to the government three months ago but it has not been published. According to a report in the Times, which received a leaked copy of the full guidance, it states that single-sex spaces should be open only to people of the same biological sex.

This would mean that transgender people would not be allowed to use toilets of the gender they live as. Trans rights campaigners say the guidance would effectively exclude them from much of public life.

Stephenson, who became the chair of the EHRC this month, told the BBC that “things could be sorted out if there is goodwill and recognition that everybody has rights”.

“There are circumstances where it is very important for some women in particular that they do have access to single-sex spaces,” she added. “It is also important that trans people don’t have a situation where there are no services that they can use.

“Nobody is expecting there to be a toilet police but equally if there are situations where there are complaints about regular problems, then people might need to improve signage, improve explanations, make sure they’ve got alternative provisions.”

Campaigners have raised concerns that the interim guidance would put the onus on businesses to police whether people were using a bathroom that corresponded with their sex at birth.

Asked whether trans women were women, Stephenson said the supreme court had ruled that under equalities law, women and men were defined based on their biological sex at birth.

But she added that “in most social situations I would want to treat people the way they want to be treated”.

Her appointment was greeted with hostility by some trans rights campaigners, in part because she donated money to the high-profile case of Allison Bailey, a lawyer who won part of a tribunal claim that she was discriminated against because of her gender-critical views.

Stephenson said she donated to Bailey’s case because she was frustrated that “women were being harassed and losing their jobs on the basis of lawfully held beliefs”.

The Guardian reported last month on internal concerns at the EHRC that the impasse over guidance on single-sex spaces was a distraction from other pressing issues including the rise of the far right.

Asked whether she was concerned that political discourse was worsening race-based discrimination and harassment, Stephenson replied: “Yes, I think so. If you look at the level of racist harassment, the level of discrimination that black or Asian or other ethnic minority people can face in the UK, we do have a problem.

“Public discourse can make that worse … I don’t know if we’re going backwards, but I can understand the fear that we might do. Sometimes it’s easy to get complacent, to think certain battles have been won, and that we’ve solved this problem.”

She added that “history shows us that rights can improve and that they can get worse” and that “the critical thing is to remain vigilant”.