I work in the legal profession and it has caused me to become very careful about what I put on the internet. It can and will make a difference to future employers. They will google you, and it can (perhaps in some cases subconsciously) affect how you're perceived if not in some cases impact whether you are hired. It can be a positive as well as a negative, but you'll never know that in advance, and once it's out there it's very hard to take it down. You have to make the choice, but I agree with WriterDude - tread very carefully.
I mean, yeah. IANAL and you are, so I won't contradict this. Prudence is good.
The basis of that prudence is knowing how to use web tools well, though. Not using them at all also sends a bad signal. I've occasionally hired folks for small projects and if I can't find a web presence of
some kind, be it a portfolio or LinkedIn or Twitter or something, I get a little concerned. It's good professional practice to have at least a landing point.
That way you control the narrative. There is something online about almost everyone, whether we know it or not. I'm pretty sure, for instance, I've got a couple of videos out there from soccer parents proclaiming me the Worst. Ref. Ever.!!!11!!!11!! I can only hope the production quality is good. Meanwhile, anyone who Googles me is much more likely to find my Twitter or my portfolio--both of which I keep properly curated for profession-related stuff.