Having an accurate and large knowledge of relevant API's will help you program faster and more accurately, and make better design decisions earlier. (You avoid the "if I had known about these API's, I'd have done it differently" situation.)
It's hard not to conclude that this is one axis in the multi-dimensional "ability" space.
Knowing APIs cuts down the time you take to implement something, and also helps you understand what and how you can achieve a goal.
Between a guy who needs to spend 5minutes googling for a class/module/function and someone who prompltly writes it down, there is no question who is a more capable programmer.
While I do not deny that memorization of APIs makes for more efficient programming, I would argue that, long term, readability, maintainability, and extensibility are far more useful time savers overall. Architecting a codebase which meets standards like these requires a different set of experience that comes with time and can't be trivially looked up online.
If I were to measure a programmer's ability, I.e., what makes a "good programmer", perhaps in the context of am interview, I would therefore not test rote knowledge.
