One reason I think some martial artists are skeptical about the value of supplementary training is that they see any kind of strength and conditioning work as detracting from skills practice in their art. Given that most of us have a fairly limited time to devote to our arts, every hour spent on S&C is an hour taken away from practicing our art. I think this idea comes from the observation that very strong, fit people are often not very good fighters, and that karateka who become overly interested in strength work sometimes let their skill practice fall by the wayside.
I have two general responses to this line of thinking.
First, the value of S&C work for the martial artist needs to be placed in a very specific context. This context rests on two points. The first point is that any person's adaption to a style of training will be inversely proportional to the time they've spent doing it. For example, if my friend Kathy, who is an experienced runner, practices running for 10 hours over the next two weeks, her time over any particular distance (let's say a mile) isn't likely to change very much. She's put in a lot of hours and miles running already. On the other hand, if I started training for distance running, using some intelligently designed protocol, I bet I could significantly reduce my mile time in 10 hours of training. I might bring my time from ten minutes to nine minutes - keep in mind I haven't run a mile all at once in ten or fifteen years. There's no way Kathy - who runs marathons fairly regularly - is going to shave a minute, or ten percent, off her best mile time over the next two weeks, month, or perhaps even the next year. Of course, her mile time is far better than mine is, far, far, better, but her rate of progress is also much lower.
This means that a weak person starting a strength training program is going to get much stronger, very quickly. An out of shape person starting a conditioning program is going to improve by leaps and bounds over the short term. Neither will end up with elite status in that quality in just a few weeks - but they will hugely improve.
The second point is that any person's fighting ability is going to be some product of their skill, strength, and conditioning. You can argue the math, whether any particular quality is more important, but given equal skill a stronger fighter is a better fighter.
So consider two very skilled, but weak and out of condition fighters. Both spend 5 hours a week training total. One devotes all of his training time to skill training while the second starts spending 25% of his training time on S&C and 25% on conditioning (remember, both start out very skilled but weak and out of shape). Who will be better after 6 months? Clearly the second fighter. The first fighter, already very skilled, will only be slightly more skilled. The second fighter will be only very slightly less skilled than the first (having spent half the time practicing his skills) but will be much, much stronger and much better conditioned - having started out weak and out of shape.
The trick is to think of training in terms of "bang for the buck" - this expression is used in this context by Robb Wolf all the time. For a weak person you can't get more return on your investment of work and/or time than by strength training. If an Olympic weightlifter comes into your gym to learn to fight, they're going to do best by focusing entirely on skill training. If a powerlifter comes in, they probably need mostly skill and endurance work. You can make the greatest overall gains by spending time on what you're weakest at.
Of course, if you want an elite level of skill, you're going to have to continually train your skills. But you're going to be much better as an overall fighter if you spend some time on the S&C side as well, because for a minimal investment of time you can make big improvements. But remember that this only works to a point. Eventually, adding in more strength training will only result in marginal improvements in strength. At that point those extra hours might be better put into pursuing greater skill.
The second overall reason to incorporate S&C work into your practice, in addition to the fact that it will make you better at your art, is for health. Being very skilled at karate won't protect you from a heart attack or stroke. Having a 500 lb. deadlift, however, will do both. Unless you are faced with serious violence on a regular basis you're probably better off doing a little extra S&C work, even at the expense of some skill work, to cover your bases. You're not going to be a very good fighter if you're dead.
Take home: do at least enough strength and conditioning specific work to take advantage of the big changes that occur when someone starts a style of training, but not enough to become an advanced strength athlete or an advanced endurance athlete (unless that's your goal, in which case go ahead).
For me that amounts to maybe an hour a week of pure strength training and maybe a half hour a week of pure conditioning training, and the rest of my time devoted to skill development.