Alkaline Water contain caustic hydroxides (OH), which also happen to some of the strongest free radicals known. As an example Drano is potassium hydroxide and lye is sodium hydroxide. You would not drink these would you? These hydroxides will severely burn the tissues. Alkaline Waters are just more dilute versions of the above.
In addition the Alkaline Waters will neutralize stomach acid leading to decreased absorption of vitamins and minerals, improper digestion of proteins that leads to an increased risk of allergies, and increases the risk of infections since the acid is there in part to protect us from pathogens.
Over alkalinizing the blood can also cause minerals to precipitate out of solution. This could theoretically cause kidney stones.
As far as I am concerned the whole Alkaline Water thing is nothing but a money making scam.