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.
As long as you didn't put him into TOS. One conversation with Kirk and the poor dog would self-destruct.
He'd have served as an excellent means of exploring the rights of non-anthropomorphic artificial life forms ... much better than those ghastly exocomps featured in "The Quality of Life". But there's something a bit too campy about K-9 that would make it a bit of a mis-match. Plus, back in the 1980s, Battlestar:Galactica's daggit was probably too fresh a memory for anyone on Trek to consider bringing another robotic dog to science fiction television.
Torchwood should have explored the sexual implications of having a fully functional robot dog around.
^ You should be ashamed of making me have bad thoughts about K-9's nose. Edit: actually thinking about it I guess that's his eye that telescopes...
Simple solution: Bring in Marvin the paranoid android from Hitchhiker's Guide to handle Kirk. Then Kirk would self-destruct.
Oh I'm hilarious! Yeah, put K9 with Janeway, the two most annoying characters in television hostory, and two characters with absolutely rediculess voices, oh and don't forget that doctor on Voyager too, he has a stupid voice. Finaly, the weird and annoying trio is complete!