How important is to maintain an online/digital presence for a graduate student?
Yes, it hurts your chances. It does not make you unemployable, far from it (I have recruited post-docs without web presence): when you apply for a position, your application file will include all the information that should be necessary for the hiring committee to decide, so they do not strictly need to see your webpage. As long as you appear on the departmental or group-level web listing of researchers… (otherwise it just looks weird)

However, look at the bigger picture: you're trying to make yourself a name in academia, so why do something that prevents it to some extent? There are good reasons for people to look up your webpage:


They saw you name, and wondered who you are, where you are, what you do.
They want to see what you have recently published, and find it easier to have it grouped on your web page that through a citation database (where your name may not be unique)
They know you, and thought about you for an open (or soon-to-be-open) position, and they want to check out if you have moved, if you recently accepted a position, etc.


So, there are two things that are absolutely needed:


People need to find that minimum amount of information about you when they search your name, and possibly some research-related keywords. A personal webpage, even a terse one, may be better for that than your departmental web page.
Your list of publications




As for myself, I maintained a webpage composed of exactly 1 HTML file, including:


my name
status (“PhD student at South Laponia U.”)
research keywords, link to my publication list (ResearcherID)
link to my CV (so I lied, I have a second file on that website: my CV in PDF format)
email address
link to my departmental website.


That's it. (It used to host my Master's thesis, but it is now hosted on an open archive, with a link from my publications list.) No picture of me or my pets, no blog/tumblr/twitter, no social websites.