The police certainly could set up a station anywhere they please- but I think that would be a very, very bad idea. Turning a city into a police state, NYC-style, certainly "reduces crime" on paper but comes at a heavy cost. Lots of people will be arrested -- and worse -- for very minor non-violent offenses and a general atmosphere of distrust and resentment will develop as the police force inevitably places the profits of business owners and their investors and the comfort of tourists over the happiness, health and even the lives of the poor, young or homeless.

I'd like to say this isn't the case but in every city I've lived with a population over 50,000 or so and a large police force, the police end up corrupt, alienated and brutal -- abusing the very people they are supposed to protect. If the woman who died in Lake Place park was homeless, a police station would have made things worse. As Vicarious observed, Lake Place park is a "containment zone" -- homeless, transients and youth are allowed to be there because it keeps them out of sight of tourists and upper-middle-class shoppers downtown and especially Canal Park. When the police start occupying the few public spaces where people with no financial means/alternative lifestyles are tolerated, that's when the arrests and beatings go through the roof. 

These are personal observations from staying in cities large and small in 26 states as well as first-hand accounts from people of many castes. The string of violence is troubling to me but be careful what you wish for. Criminals aren't the only ones who beat, rob and kidnap for questionable reasons.