Why is it not dangerous to dissolve NaCl?
[A really-good answer to this question should include quantitative evidence, but here is my hand-waving version...]

Solvated sodium ions ($\ce{Na+}$) aren't all that reactive.  Surrounded by a lot of polar water molecules, they're actually quite stable.  What you're thinking of is solid sodium (Na) which will "react violently with water" until it's all dissolved - dissolved that is, into stable sodium ions.

Same kind of logic applies to chlorine.  Solvated chloride ions ($\ce{Cl-}$) are stable and not nearly as poisonous as chlorine gas. (Note:  everything is poisonous at some level, i.e. "the poison is in the dose".  In saying chlorine gas is more poison, that really just means it is a more 'potent' poison:  it will kill at lower doses compared to chloride ions.)



Another way to think of it: 
Solid sodium and chlorine gas are at high energy states.  Since they can reach much lower energy states by ionzing in the presence  with water, the WILL break down and ionize in the presence of water