Nobody loves K9 more than me (hell, I built my own), though now as an adult, looking back at his stories he was obviously a gimmick at least in some cases, and caused more problems than he was worth.
Thinking about it, K9 would have made much, much more sense in an iteration of Trek.
Imagine if Data had built a K9. K9 would be awesome on the Enterprise. For a start... no stairs (well, he might have a bit of trouble in Ten Forward). Lifts for him to go wherever he needs to go. Nose Laser for fending of invaders.
Not to mention he's a very powerful mobile computer. He would compliment the Enterprise Computer.
And no ethical dilemmas or conundrums. The Doctor never had any qualms or moral episodes where he wondered exactly what K9 was (dog, or robot? Tool, or friend?), unlike Data or those gay exocomps (I can say that because I'm gay). He would be a handy character to have around, without all the complications.
Of course he would fit in just as well in other shows as well. He'd work very well with O'Brien on DS9, and maybe even on Voyager. Hell, have him on each... K9 was easy to rebuild and clone (there were four different versions we know of) so no reason every ship/station can't have at least one of him.