Given how popular Klingons seem to be with the fan base, does anyone know why no one ever considered a TV series based around a Klingon star ship? That's what I was really hoping for when I first heard rumors of Voyager.
In short, the numbers were dwindling through the three series that started in the 1980s and the point of making a TV show is to make money, so when you already have a limited audience, limiting it even more by appealing to only one segment of your fanbase (which is smaller than your viewer base) is unprofitable and unwise.
If you want a more in depth reasoning, then read on:
While this isn't canon (it can't be, we're talking behind the scenes), and I can't cite a specific source, this comes from a few discussions I had in pitch sessions with someone at Star Trek: The Next Generation and what I've read elsewhere along the way.
One big issue that developed after ST:TNG was a limited audience. People began to think, "Oh, another Star Trek series. Well, I never saw the last one, so I'll have no idea what is going on with this one." There was also an issue with an aging audience. People were thinking of Star Trek as something for the old folks who remembered the original. While this wasn't as big an issue at the start of ST:TNG, it did become an issue during Star Trek: Deep Space Nine and Star Trek: Voyager. Demographic ratings showed an audience shifting past prime (in terms of preferred targeted age groups).
The rationale behind doing Enterprise as they did was to make it clear it was a fresh start - going back before 24 years of accumulated TV history and creating a series that anyone could start watching without having watched all the others. That was also one of the reasons for dropping "Star Trek" from the title.
The series Enterprise did not reach enough of a new audience to make it viable for the full 7 years the other newer shows lasted. It took the reboot done on screen a few years ago, a reboot that was advertised and publicized as a reboot, to catch on.
Now, you've probably started saying, "But this doesn't address Klingons." Not yet, but here's the issue: If each Trek series was drawing a smaller and smaller audience because the audience was aging and new (as well as younger) viewers were just not coming in to the fold and joining the world of Trekdom, think about the production angle.
Your demographics are already shrinking. So what would doing a Klingon series do? Well, the Klingon fanbase would love it and there are Trek viewers (not Trekkies/Trekkers, but just viewers) who would watch it, but would the average Aunt in Peoria (as Gene Roddenberry put it sometimes) watch it?
While the fanbase of Klingons, within Trekdom, may be large, the viewer base of Trek on TV was getting smaller. At that point, doing a series that focuses solely on Klingons, which would excite Klingon fans but would turn off other Trek fans, and would appeal to even fewer overall viewers than any broad-based Trek show would.
Even if you assume that there would be an audience for such a series (which I personally think is a bit of a stretch, as Klingons do not make up a significant portion of the TV viewing audience), the production costs and complications would be prohibitive.
The lack of working Klingon actors in Hollywood means that full prosthetic makeup would be required each and every day for the majority of the cast. Not only would this be expensive and difficult for the actors, the amount of time it generally takes to apply such makeup would probably result in much longer than typical shooting schedules for each episode, which would in turn compound the additional production expense.