There are two major types of advisor approaches which I've encountered during my studies:
Hands-off, abstract guidance only. Hint at which areas are interesting or currently active, but do not throw yourself into your student's work. Usually such advisors will gladly proofread the student's work for errors, but they rarely know what to do if a student gets seriously stuck. Advisor sessions may be spent discussing current trends, future work of either the advisor or the student or, in some cases, simply "life advice".
Co-worker approach. The advisor treats his student as an equal and tries to immerse himself into the student's work (or vice versa). By having a lot of exposure to the advisor's routine (how he goes around solving problems, how much time he spends doing research, how he prepares for lectures, etc.), the teacher aims to guide the student by example.
These two approaches are often seen in combination (while the student may be treated as a coworker, he still is usually the one to do the typesetting, the programming, and other time-consuming tasks).
Personally, I prefer the second approach, which is likely because I am not that settled in the "theoretical research" routine. Some of my friends, who have known since their Master's studies what they want to do, are very enthusiastic about having a well known professor with the hands-off approach.
To answer the question about either approach fostering independence, I believe the hands-off approach leads to much larger independence as a researcher, but the coworker approach may lead to a much better work ethic (mirroring the professor), which is tremendously important in science.
It seems to me that students have a harder time successfully completing research (asking the right question, having the right insights, etc.) than being independent. Therefore I believe that student/advisor relationship does not need to promote "independence" in any substantial way.
