There are plenty of waterpipe cafes (nargile in Turkish) that provide free internet. Especially the ones in Besiktas bazaar area seemed to have good connectivity.
Other options are the fast food restaurants like Macdonalds or Burger King.
simdi cafe, on asmali mescit, in tunel is great for coffee and snacks. and free wireless.
it is just off istiklal caddesi - near tunel in the beyogly district.
also all house cafe's offer free wireless.