We simply would not have a good idea what to look for if looking for non-water-based life.
We know that life using aqueous chemistry can occupy a wide range of temperatures, pH values, light and dark, using photosynthesis, chemosynthesis - even microbes that "eat and poo electricity" have been discovered in a wide range of environments. (Despite the cute title, the article is a good review of the subject - bacteria and other microbes that derive their metabolic energy directly from external chemical potentials.)
This large variety gives us a sound basis for choosing signs or markers of life processes. Things we should look for that would suggest life.
Since we have no examples of life processes without water, and not even any good theories of what it might be like, it is much harder to propose what signs or markers are worth spending $10,000,000 or 100,000,000 or more for a space probe to look for it.
SETI - the Search for ExtraTerrestrial Intelligence - looks for signs of intelligence, with no bias on chemistry. They certainly don't have a bias towards water, and actually they don't even have a bias towards life. Just intelligence.
Looking for water-based life is a little bit of a case of looking for your keys under the streetlight because that's where the light is, but at least here you know you have lost at least some of your keys under the light.