I'd like to give some high tech answer, but that's not the case. If you interface externally - consulting, dev evangelism, conferences, tech sales - then yep, paper biz cards are still used.
WTF? Are they not professionals? Of course they use business cards!
I've never really had a need for them in ~10 years. Mostly dealing with in-house clients though. That's why I was wondering.
Freelancers, contractors, consultants - yes. It's the easiest way to give email address with such low-tech solution.
For pre-sales calls: Yes.
For on-site support calls: NEVER! - let them call the support line again next time.