As I battled a pushchair across the unhelpful Amsterdam cobbles, three young Englishmen approached. “Where’s the red-light district?” they asked this bleary-eyed, breastfeeding mother. I thought about directing them straight into a cold canal, but politely waved in the direction of what the Dutch call De Wallen.

A decade ago, I had no idea that they were the first drips in a flood of tourists. Now, 17 million visitors a year stay in Amsterdam, many keen to ogle women in red-lit brothel windows, party, and turn the centre into what city ombudsman Arre Zuurmond has called an “urban jungle”.

In some ways, it’s nothing new. Since the early days of seafaring, sailors have been stopping off in the port of Amsterdam to avail themselves of the remunerated charms of its women – as witnessed by the jaunty 17th-century sea shanty The Maid of Amsterdam, in which a man pledges to “go no more a-rovin’” with a red-lipped dame who has been his “ruin”.

Roll forward a few centuries, though, and there’s a widespread feeling that the sex district – or rather the visitors it attracts – is proving the ruin of Amsterdam. This week, after months of hinting at action, Amsterdam’s first female mayor, Femke Halsema, has revealed four potential futures for the famous prostitution windows, including shutting all 330 of them and moving the sex district elsewhere – possibly to a “sex work hotel”. Keeping the status quo, for the city council and many of its residents, is not an option.

Living in a city where your normal (cycle) path might take you past women selling themselves in windows is an odd thing. Suddenly, there’s something terribly interesting on the other side of the street to point out to the children (alongside a resolution to take the long route next time). On the bright side, the sex workers generally look bored and grumpy rather than X-rated, and the Victoria’s Secret billboards at Schiphol airport are probably more revealing.

Yesterday evening, pedalling through the medieval district, the narrow, thronging streets felt a bit like the start of a college ball. Window brothels lining the Oudezijds Achterburgwal were either empty or the red curtains were pulled shut. But there were also signs that the city was watching: a few police on horseback, a huddle of city wardens reminding tourists to “enjoy and respect” Amsterdam, and adverts informing miscreants that street drinking or peeing risk a fine of up to €140 (£125).

‘Amsterdam’s first female mayor, Femke Halsema, has revealed four potential futures for the famous prostitution windows.’ Photograph: Piroschka van de Wouw/Reuters

The atmosphere isn’t what you’d call homely, but normal life in Amsterdam doesn’t bring you into contact with sex workers unless you look them up. It’s a different story with those tourists who throng in front of the windows, leering without buying, eating cheap food on residents’ front steps, spilling over the city and generally being an obnoxious nuisance. It’s obvious that something needs to change to let this group know that they are not welcome.

The Dutch approach to prostitution is largely practical: sex work will always exist, so better for everyone to legalise, control and tax it. Sex workers are registered with the chamber of commerce and the city council – and must have health insurance and regular checks. This makes a lot of sense to a feminist, and Foxxy Angel, of the Proud Dutch union for sex workers, says the window structure gives them safety and security – as well as the chance to look a client in the eye before saying yes and opening the door.

But the council has so far failed to clean up the problematic aspects of the red-light district – fraud, money laundering and human trafficking. In 2007 the city started Project 1012 to address widespread criminality: however, analysis by the city audit office last year found that while 112 windows and 48 coffeeshops – where cannabis is smoked – were closed, the project did not succeed in breaking down criminal infrastructure or bring in any economic upswing. From 2005 to 2016, the report said, there were at least 119 victims of human trafficking.

My Red Light, a project that invited sex workers to run their own brothel in city-owned premises, has had mixed results too, with financial difficulties, concerns about human trafficking and complaints from sex workers about heavy-handed enforcement.

Some people feel brothels are part of the charm of the beautiful De Wallen, but it seems obvious that in an internet age where a screen is as good an advertising space as a window, they could easily be located elsewhere. Angel tells me that many sex workers who visit the Proud office in the red-light district are concerned about the future but admit that a sex hotel-type structure could offer the same kinds of protections. Lyle Muns, spokesman for My Red Light, says sex workers certainly need a good alternative if the walls go up in De Wallen.

Finding another location would be a problem in a city with a massive space and housing shortage, but even talking about moving prostitution from the red-light district is a good thing. If it flashes up a big, red “stop” sign for those thoughtless, boorish tourists, or diverts them somewhere else, many Amsterdammers will feel a profound sense of relief.

• Senay Boztas is an Amsterdam-based journalist who writes on Europe, particularly the Netherlands and Belgium