Boycotting bottled water means you support the idea that public access to clean, safe water is not only a basic human right, but that it's a goddamn technological triumph worth protecting. It means you believe that ensuring public access to this resource is the only way to guarantee it will be around in a few more years.
Clean, safe drinking water that flows freely out of our faucets is a feat of engineering that humans have been been perfecting for two millennia. It is a cornerstone of civilization. It is what our cities are built upon. And over the years the scientists and hydrologists and technicians who help get water to our houses have also become our environmental stewards, our infrastructural watchdogs, our urban visionaries. Drinking the water these people supply to our homes is the best possible way to protect future access to water worldwide.
Companies that package water in a single-use bottle are not concerned with the future. They are not invested in the long-term effects of climate change on an endangered watershed, nor are they working to prepare a megacity for an inevitable natural disaster. What they are interested in is their bottom line: Marketing a "healthy" product to compensate for the fact that people are buying less of their other products that are known to case obesity and diabetes -- and selling it for at prices that are 240 to 10,000 times higher than what you pay for tap water.
The reason you should boycott bottled water is because it enables a bullshit, backwards vision for society.