I know that in most of Asia drinking water was boiled (making tea and the like), and in much of Europe alcohol was used, but what did people in the Americas use before European settlement to keep their drinking water safe to drink?
In the U.S. Southwest a number of naturally purified sources are available thanks to the geology. For instance, the cliff-side villages of Ancestral Puebloan people of the Mesa Verde region in southwestern Colorado had access to naturally purified seep-springs. The sandstone formation (the erosion of which created the alcoves in which these villages were built) of Mesa Verde creates natural seep-springs from the percolation of water down through the mesa-top into these alcoves in a process of natural filtration. We know that Ancestral Puebloan people who lived in these villages took advantage of these seep-springs in several villages because of man-made alterations to the rock surrounding the seep-springs in order to better collect and retain the water. For instance, the circular depression in the middle of this pool formed by this seep-spring at the back of Long House.
Similar springs are common, though perhaps not as obviously related to human habitation, in other areas of the Southwest with similar geology.
Additionally, taking advantage of rainwater for drinking is another effective strategy. Since growing corn in the arid Southwest already requires using supplementary water in addition to rainfall, one strategy was to collect rainfall from areas that were not agricultural fields to redirect onto those fields. Often this was done via check-damns built in existing washes to create small reservoirs. These washes run dry most of the year, but during heavy storms they tend to flash-flood, at which point the check dams would help collect some of that floodwater for use on fields primarily, but also as drinking water. Of course considerable sediment tends to collect in this floodwater (and potentially contaminates) so it is more effective as a way to water agricultural fields than providing clean drinking water. Ceramic pots may have been used to help collect rainfall (since they were used to store water), but that is speculative rather than archaeologically evidenced.
Edit: Small corrections.
Just to nitpick, the Aztecs did not live in South America, but in North and Central America. The Incas were in South America.
Minor point, but your assertion that "in much of Europe alcohol was used" is incorrect. You may be interested in this section of our FAQ, on drinking water.
I'll add - much of the reason that drinking water is boiled in Asia is because of intensive agricultural practices and widespread use of various types of fertilizers. If you're fertilizing your crops with human manure, then getting your drinking water from the creek that flows through, you'll need to boil your water. The denser a population, the more important this becomes. Much of North America was not agricultural, and as a result in general water was cleaner. That said, as a kid growing up rurally I was taught a number of rules regarding water in the woods -
i.e. avoid poisons, and get as close to a spring as possible. We got our water from a spring. The main thing people did in large parts of the continent was not to purify the water, but to obtain it from clean sources.
