Unsatisfying answer: it depends what you mean by "affect". Different versions of this theory have been known as the Sapir-Whorf hypothesis, or just the Whorf hypothesis, or linguistic relativism, or linguistic determinism.
At one extreme, you could ask if language constrains what we can think, if we're unable to think of things that our language does not allow. This version is common in pop culture (like Nineteen Eighty-Four), but has been pretty conclusively debunked.
At the other extreme, you could ask if language determines what words we use for things. But this version is so weak as to be meaningless: of course what words we've heard affect what words we use.
The more interesting part of the question lies in between these two extremes, but as jk says, this is a very difficult topic to study because of how difficult it is to make solid, falsifiable hypotheses. Famously, Kay and Berlin showed that the way a language categorizes colors affects how quickly and easily people can distinguish them. Native speakers of English, which fundamentally separates "pink" and "red" and "orange", find it easier to distinguish reddish colors than native speakers of Swahili (which doesn't). But we find it harder to distinguish bluish colors than native speakers of Russian, which fundamentally separates "indigo" from "cyan".
Now, this doesn't mean English-speakers can't distinguish indigo from cyan. But there's a definite, quantifiable effect that can be measured in the lab, showing that our language to some extent affects the categories we impose on the continuous world, and we're faster and more accurate at identifying differences between categories than within them.
What does this mean for, well…anything except colors? Unclear. If you look into linguistic relativism you'll find a huge amount of discussion and not a lot of hard data. I would consider this one of the big questions of the field of psycholinguistics.