Well the amount of a substance that can dissolve in water is largely inpart with the size of the solute and the polarity of the solute (determined by electronegativity). When NaCl is mixed with water, a portion of the NaCl disassociates into both Na+ and Cl- ions. These ions then become
saturated with the H3O+ and OH- ions from water (or you can acknowledge them as H+ and OH- ions). These ions surround the Na+ and Cl- ions. Na+ is attracted to being surrounded by the negative ions (anions) while Cl is attracted to being surrounded by positive ions (cations). Because NaCl is polar when mixed with water, this happens relatively easy. And because the ions are very simple and small, you have a larger surface area for dissolution to occur. I hope that helps
smile !
At a point though, the solution of NaCl and H2O will become supersaturated in accordance to the temperature, and no more NaCl can be dissolved into it due to the equilibria reached between the formation of ions from H2O and the formation of ions from NaCl. At greater temperatures, this equilibria is shifted to be able to dissolve more NaCl (plus more energy is provided to split these compounds into their ions). However, precipitate will form if the temperature is lowered once again.
Though with your main question, even though salt water is not super warm, I believe that the amount of salt it has is less than the supersaturated point of pure water, so precipitate doesn't form. However, I am not entirely sure if those places are supersaturated with NaCl. If it is though, the precipitate formed will sink to the ocean floor. Also, there are probably a lot of polar impurities in the water aiding in NaCl dissolution, plus I'm sure a lot of organisms rely on high water salinity for survival, and hence, use the NaCl for various purposes.
Once again, water does have it's limit to how much NaCl is can dissolve per volume of water and temperature. Those places likely haven't reached that limit yet ^^.