You ask about bringing a copy, but the real question is about using it during the interview. Having a digital copy in your pocket/bag is a reasonable precaution, but they won't even know unless you want to consult it.
I've encountered two types of interview situations where API knowledge is relevant. One is when they're just asking questions to assess technical knowledge (e.g. what's the difference between an X and a Y). In those cases the whole point is to see what's in your head; if you don't know you can say "I don't know but can look it up", but they probably won't want you to. Probably they'll just move on to the next question.
The other situation is when you're writing code during the interview. When I conduct interviews of this type I don't design the problems to rely on arcane knowledge. Occasionally during an interview somebody will say "I don't remember the exact syntax for this", where "this" is a minor detail compared to the problem. In my experience interviews usually just ignore that; after all, as you said, we assume that in a real setting you'll look it up. If that happens to you you could say something like "I don't remember the exact syntax" while you pull out your phone, demonstrating that you're prepared to address the problem. Whether the interviewer wants you to proceed or just tells you not to worry will probably depend on time.
All of this is for small details, though. If you are, say, applying for a web-development job and you say you don't remember the syntax of PUT or what a return code of 404 means, being able to look it up isn't going to matter. Basic knowledge -- whatever is considered basic for the particular position -- they'll want to see in your head.
So, bottom line, make sure you know the key APIs, it doesn't hurt to have the rest in your pocket, and play it by ear during code-writing interviews.