Why do most organisms depend on water?
One source says,


  Reactions cannot happen without water molecules.


It's not the most impressive source, but here's another source that has the same theme, with more details:

Cosmo Quest Forums


  The water molecule forms non-covalent bondings. It is bipolar and
  easily 'gets inbetween' other molecules which want to marry by charing
  their electrons more permanently. This makes water a great dating site
  where for example aminoacids quickly can try out their fit to many
  different parts of an RNA molecule. If they don't make good enough
  covalent bondings, water will immediately be there to separate them
  until the next one comes along for a try.
  
  Methane (on Titan) is as liquid as water and does transport stuff
  around. But it lacks the ability to interfer with how other molecules
  bond. Biochemical processes would be much slower in methane liquid
  than in liquid water.


Water is also a fabulous solvent; so many things can be dissolved in it. As others have mentioned, the simple abundance of water in the universe may be a factor.

However, I don't know anything about H2S' bonding ability or its power as a solvent.