Below grade spaces always have the possibility of flooding. Given you are in a brownstone, I suspect the only way to do the basement would be an interior wall french drain system, I had one put in 18 years ago and the only floods I have had were when we lost power during one of the hurricanes (I now have a generator for the pump), or once when the pump itself burned out due to the outlet getting frozen over in the winter (my stupidity, the pump itself has a lifetime warranty). It is as others describe, they dig up the concrete on the periphery of the basement, dig down, they tap into the walls to releive hydrostatic pressure, then put in the perforated pipe and gravel, and this is led to a sump. In NYC the gutters are fed into the storm sewer system, so it may be legal to feed the pump output to this (I don't know for sure, though, the contractor would know). It should stop the water you are seeing. I would also seal any cracks in the wall with hydrostatic cement and paint the walls and floor with waterproofing paint like Drylok.
As far as finishing off the basement, it does pose challenges because it tends to be humid down there. If you are going to finish it off, it is important that insulation doesn't touch the outside walls (using foam on the actual wall then putting insulation is one way around this, or build the framing for the sheetrock off of the actual wall so the insulation won't touch). There is sheetrock now that is mold and mildew resistant (course yours truly didn't use it) that can help.
The real key is keeping the basement as dry as possible, dehumidifiers or an air conditioner will work. The Wave system is an option, it basically is a vent fan tied to a humidistat, that creates negative pressure in the basement that draws airs via vents from upper floors (that tend to be drier), venting to the outside. It does exchange the air, but it also draws heated or cooled air from upstairs and vents it outside, so it does cause your heating or cooling to work a bit more. The other option is they have vent units that take air out of the basement while pumping in air from the outside, and through a heat exchanger heat/cool the air from outside with the air expelled from inside, which brings in fresh air , vents 'old air', and the heat loss is minimal (on the other hand, it also can mean bringing in more humid air from outside). With the wave system, one problem may be if you have things like hot water heaters or a furnace in the basement, in creating negative pressure it could suck CO back into the house, so you have to be careful with that. My basement is pretty small, and a dehumidifier does a decent job. In theory you could create a wave system out of a strong bathroom ventilator, a humidistat and opening some vents up to the upper floor from the basement for a fraction of the cost of a wave system.
The safest route in the basement is not to finish it off, but it can be done if precautions are taken, you won't necessarily have problems with mold and such is you do it correctly (or have it done). It is a lot nicer to have it finished off, makes it a bit more pleasant to be there, but at the very least paint the walls a cheerful color and on the floor, put down rubber tiles that a)are pretty impervious to water and b)are soft on the feet
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In the end, others are right, basements are at risk from broken pipes and other unexpected disasters, and one of the biggest things is to have things out of harms way, have things off the floor, whether it be the trains, shelves, or things like washers/dryers, hot water heaters, etc.