Which phrase is more common in AmE:
What do you do to kill time?
What do you do to pass the time?
Are both of them equally common?
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This says nothing about spoken language, of course, though it can be indicative. I don't know of any spoken American English corpora that I could look this up in.
However, the meanings of the two phrases are not precisely the same. There are many cases where you could use either without much practical difference, and in such cases you can use whichever. They have quite different nuance, however. Killing time suggests that the activity you use to "kill time" isn't particularly worth doing, while doing something to pass the time carries that connotation less strongly. If one is killing time, there's a suggestion that you resent having to do so. Overall, there's more of a suggestion of irritation and pointlessness to killing time than passing the time.
Also, some instances of "pass the time" may be "pass the time of day", though not very many will be. That has a quite separate meaning.
Ngram shows "pass the time" is more frequently used but a gentle reminder, ngram is mostly based on books not spoken language.