Should student or supervisor be corresponding author for publications based on student research?

"The corresponding author is the one who take the responsibility of a paper". I've never heard this before. For example: http://sciencecareers.sciencemag.org/career_magazine/previous_issues/articles/2010_04_16/caredit.a1000039 says "The corresponding author is the point of contact for editors, readers, and outside researchers who have questions about the contents of the paper. Often, the corresponding author is also the last author, but she or he may be listed first or even in the middle of the author list."

All authors take responsibility for the paper (or should). The point of the corresponding author is who to contact if you want to correspond about the paper. If this were someone who was likely to move institution (because they are finishing, or have finished their study), they are going to be hard to contact, so make it someone who's likely to hang around for a while. I've never seen anyone take any notice of who the corresponding author is.

I was always the corresponding author; my advisor(s) thought it was good for me, and they had other things to do than to fiddle around with LaTeX...

So, to answer your question, I think it is good for phd students to be the corresponding author; besides, if there is any trouble, you have always your advisor/coauthor to ask.

Since the ordering of authors differs between fields the meaning and usefulness of a corresponding author also varies. In fields I am familiar with, the corresponding author is usually the same as the "first author" (quotes because it may not be literally the first). Many journals therefore do not explicitly identify a first author unless different from the "first". There are then several cases where the corresponding author may need to be identified. One example is when a person lacking a permanent academic address is first author. Then the supervisor may take on the responsibility for the paper and be corresponding author. This can be important since it can be near impossible to track down someone who has left academia and so the supervisor stands for continuity in terms of contact. There are many variants on this and in some cases, a person heading a project or who by legal obligations carries responsibility for a project may be identified as corresponding author. This could be the case with some governmental organisations where communications are funnelled through hierarchies for bureaucratic reasons. I am sure there are lots of examples good and bad but the main purpose of identifying corresponding author, unless first, is so that anyone requiring more information can go directly to the main source for such.

So based on this background and the field you are in you may find a good way to determine corresponding author. In most cases, I would say it is the person who has done the most work, or the one who "owns" the project. It is not clear in some cases whether it is the student or the advisor who should be corresponding author. One also has to weigh in the intellectual work behind the project as a whole and from that perspective the person who has done the work, perhaps a detail in a much bigger perspective, may not be the appropriate person for details although that person has done most of the work for the paper in question. So in some cases the question is definitely harder to answer. Not being corresponding author, does not necessarily detract much from being first author since such details are not visible in literature searches and CVs.