James Greenhalgh, 28, had to “beg and borrow” to build up the £35,000 needed to start Flamingos Coffee House in the Central Arcade off Briggate.
But since opening late last year, he says nearby drug use and threats towards him and his staff have become worse.
Mr Greenhalgh said: “It’s been progressively worse and worse – people smoking spice and taking drugs. Shop owners are getting threatened, having to close our doors because we are getting threatened.
“Our takings just drop and we are really struggling.”
He claimed that a recent incident involved a group of people banging on his cafe windows and shouting at staff, and said previously the shop was egged.
The cafe closes at 9pm and community groups make use of it, but when anti-social behaviour occurs it forces Mr Greenhalgh to shut his doors early. He added that during one incident, a customer said to him: “I want my money back, I want to leave. This is awful.”
In October, a multi-agency Street Support Team was launched, and part of its remit is to deal with anti-social behaviour. But the former civil servant, who employs four people, would like to see more action taken against the culprits, who he believes are the same 20 to 30 people every time.
Mr Greenhalgh said: “I needed a change of scene and always wanted my own business and I’m doing it.
“It’s doing well but when we have the anti-social behaviour we have to shut the shop earlier.
“It’s killing us off.”
Inspector Andy Berriman, who heads the City Neighbourhood Policing Team, said: “Officers from my team have been to see the business owner and I have offered to meet with him to discuss his concerns in detail as we are keen to address the issues raised as quickly and effectively as possible.
“We have increased our patrols of the area and I have also authorised a dispersal order specifically to address the problems he [Mr Greenhalgh] is referring to.
“This will allow officers to effectively ban those causing the issues from the city centre for 48 hours and should they return during that time will face arrest for a criminal offence of breaching the order.”
He said the Street Support Team and other groups are focusing on the issue."