When doing stage combat, safety is a large part of it. All strikes do not hit or are exaggerated, and many of the more intense moves are controlled by the victim.
What would be a good approach to adapting Karate moves to stage combat?
Stage combat is multidisciplinary. You have to know both enough of the martial art or fighting style you are trying to emulate, and you have to understand the restrictions of the stage, and the audience view point. You use these things to hide your cheats, and your non-contact fighting.
The details of the different aspects vary based on whether the staged fighting is live or filmed. For example, you can use different camera angles and lighting to substitute a body double who is more skilled than the main actor. Additionally, with filmed fight scenes, you can break the fight into much smaller pieces which are easier to reassemble into a contiguous fight scene later.
All of this has to be done in a way that is consistent with the rest of the setting. If we are dealing with modern warfare, basing the techniques on a stage variant of Krav Maga would be a good choice. If we are dealing with something set in historic Japan, you'll want to base it on styles from that area and period of time (at least reasonably look that way).
A fight choreographer needs to have the whole sequence outlined from start to finish and know the following details:
The fight choreographer will be working with the director to get the details worked out, and where you focus on a character for any reason.
Here's the main problem:
Efficient, deadly moves are not pretty. They're mostly invisible.
Pretty, showy, artistic, theatrical moves are not efficient, or deadly. And they're very visible.
Now, from there, take an efficient, deadly, invisible move and make it pretty, show, artistic and theatrical. There's your recipe.
I'll throw you a bone: The punch does not come from the hip, it comes from WAY FAR BACK!
Now, once all this has been said, you can't "adapt Karate to stage combat". What you can do is adapt stage combat to look like karate. And that's going to depend on the guy doing the choreography. Big stances, hand movements... Lots of aikido-like throws and rolls, maybe. Who knows.
This question really belongs in some kind of Theatre Q&A, not a martial-arts Q&A, unless you need amazing detail. And you most likely don't.
Important disclaimer: Reading this answer, or any answer to this question is no substitute for getting actual training on stage combat from someone who knows what they're doing. Staged combat can be dangerous when performed incorrectly.
The elements of getting stage combat looking good are reasonably simple, although actually doing it well is another matter.
For any strike there are three elements: the build up, the impact and the follow-through.
Unskilled fighters will tend to show a much greater build up, as they telegraph their strikes, but in practice this phase will cover everything from when someone starts moving to the point of impact.
There are two things that really sell the impact - sound, and timing.
The classic technique to get sound is called the "nap", and it is where the attacker (usually) slaps their own body with their non-hitting hand. This also helps the defender time their reaction right. If the form of the martial art prevents this (i.e. karate's double punch) then the defender can do the nap instead. Other options include offstage sound effects, foot stomps (a la professional wrestling) and a shout to mask the lack of impact noise.
The other important thing about the hit is don't actually hit them. With an audience on only one side you can aim a punch to pass between the audience and the victim (especially head punches) and - as long as the defender gets their timing right - it will look realistic. Kicks are much harder, and typically will need to be pulled a little short. Roundhouse kicks will work better here than front kicks. If the defender is willing, you can make a small amount of contact with kicks, but you mustn't actually kick them - just give them a little push on the chest / shoulder with the sole of your foot (not the instep).
Weapons should be swung in such a way that a missed block will miss altogether. This can be achieved by aiming swings past the body by an inch or two, or by keeping just out of hitting range. If the margins are smallish the audience won't notice.
This includes any literal follow-through on the hit, but also includes the victims reaction. For karate this is probably very little follow through on the hit, but a big follow reaction from the victim.
You'll notice that none of the key points above actually refer to any form - just principles. You can apply these principles to fencers, pirates, soldiers, untrained brawlers and martial artists simply by adopting the relevant postures, attack / defence styles, apply the right philosophy to the choreography, and (most importantly, actually) footwork.
Sources: I've been trained in staged combat and am a professional wrestler.
Here are a few suggestion to take into consideration...