I think BlackBerry's good or, over here, basically only business reputation simply developed during their glory days. If you wanted to communicate, get and send mails instantly on the fly plus want a good battery life, BlackBerry was the way to go. Last but not least thanks to BES and device managements.
Back in 2012 I bought my first berry, a Bold 9900, mostly because of their business reputation. For me business related devices need to operate fast and efficient for mails, need to be highly reliable and feature superior PIM functionality. Last but not least I wanted a physical keyboard and, even back in first half of Q2 2012, the choice was highly limited.
Is BBRY still business? I think yes and I think they still have that reputation. But it is not always a good reputation. Obviously many workers who used BlackBerry (or were forced to use them) don't know anything about BB10, even though it is now available since more than 18 months. Too many people still think about their old OS6 Curves and compare them with newer devices. They don't get or know everything has changed, plus I often think it is kind of a hate relationship, since the BlackBerry is always THE symbol of working everywhere on the go.
Personally I have to say the keyboard alone makes a huge,difference. I've used physical keyboards on my phones since years through various manufacturers and platforms (mostly Nokia) and like the actual tactile,feedback they offer over the glass surfaces we got today. Vice versa I've used full-touch phones I got supplied for work (which is kind of ironic, the BB sitting in front of me is always my personal device, while that modern full-touch bar is always the business phone) and still dislike them. While they are sometimes faster to type with, they are far less accurate, relying on that f***Ing autocorrect can be highly annoying and is even worse when you need to use many special terms and words for work anyways.
Now I got my first full-touch device earlier this year in form of a Lumia 1020. I only it because it was a very, very attractive price, because I wanted the camera (which really is awesome btw) and O always wanted to try Windows Phone. What can I say, six months later I popped the main SIM out of it and placed it back into my Q10. Unfortunately it is gone at the moment to repair that double-typing, but I am really,looking forward getting it back. Even the modern swiping keyboards can't replace a physical keyboard IMO, especially for,longer texts. I am writing this wall of text right now on the Lumia and I don't enjoy it. I write less than normal because I can't wait getting done and end typing, it simply is annoying, that soulless glass slab is uninspiring, has no feedback and, to sum it up, doesn't "connect" me with my writings the way a physical keyboard does.
Yes, I think this is not an unimportant thing for businesses.