I think that a web presence is a must in academia (at least nowadays!).
At the very least you should have a site for the current course you are teaching, as a graduate student. This is not necessary, but it is starting to become expected by students (although, who cares what they think ;) ). I think a webpage is good because you can update it at a moment's notice, almost everyone has access to it, and if not, it's easily done, and you can also provide solutions for problems, quizzes, and past exams, as well as have links for cool math-y things (my area of expertise), wolfram alpha apps, java apps, matlab code, etc.
However, I believe that this post was inquiring more along the lines of having a webpage as an early researcher. In that regard, yes and yes (and dare I say yes again?).
Having a general page of your research interests, several sub-topics, collaborators, and even, (dare I say it?), a personal portion of it about you, is a good idea. People expect to be able to access documents for pre-prints, post-prints (assuming you have the appropriate copyright), software, CV, etc., and at this point it is not unreasonable for them to think so.
I would also suggest having a site not at your university. You can redirect your university site to your other site and you don't have to worry about migrating files over when you move from grad to post-doc to post-doc to tenure-track at a tier 1 research institution (except for maybe the last transition).
Also, if you host your own site, you can have your own diaspora pod running, and we can move more towards open-source networking.