I recommend looking up people like Dr. Susan E. Brown and her research into osteoporosis related to over acidity of the body. The body tries very hard to maintain a very narrow blood ph of about 7.4, and it will go to the lengths of even leaching alkaline forming minerals like calcium and magnesium from the body to do so, or breaking down muscle to get glutamine to make ammonia (very alkalizing).
It's the acid accumulation in the tissues which cause more immediate damage and issues, and which the body has a harder time correcting.
Of course, the stomach is meant to contain a lot of acid! But ironically enough, often in those whose bodies are chronically acidic in the tissues, and where the bones and muscles are being broken down, there is too little HCL in the stomach which leads to problems--like inefficient absorption of B12, inefficient break down of proteins (especially complex ones like in meat or dairy).
There is research out there which supports the acid alkaline theory. However, if one has already formed a belief system about it, one tends to see what one wants to.
How do I know there is something to it? At the end of the day I care less about the vaunted scientific knowledge about it, and more about observable effects in my own laboratory: my body. When I have a disciplined and more alkaline forming diet, I can be around any number of people with a cold, flu, etc. and not get anything for years. I worked at a school and was in close contact with a lot of children with various infectious pathogens and went over two years without even a hint of the common cold.
Back when I ate more "normally," or rather the standard American diet (truly sad), that would not have been the case and I experienced getting sick every few months or so on average.
Why would this be different? I believe it's due to the increased oxygenation of my body, and overall greater balance and nutrition density which leads to a stronger, healthier body and increased immunity functions.
A closed mind and unwillingness to experiment are your loss, and my gain. Sadly, too many make mainstream science their major religion. Studies and research are definitely important and of value, but so is your own experience. Neither science nor mainstream medicine know all there is to know about the human body. Experimentation and experience can help to fill in the rest.
By the way, to be more clear, I'm not talking about "alkaline water" so much as dietary alkalinity, which happens primarily through mineralization of foods. I don't have much experience with the former, but I have 12 years of experience with the latter.