It's not uncommon for many of us as martial artists to workout outside of the dojo, whether it's weight training, jogging, etc. However, many of us tend to work out the "wrong" way. Granted, in my opinion, some activity is better than no activity at all, but in many cases there are more optimal ways of working out.
Most of it depends on the particular style you study. The main mistake people tend to make is doing the same type(s) of workout no matter what style they study. Different styles have different physical requirements, and thus the types of workouts done should differ as well. The main physical attributes in play are aerobic power, anaerobic power, flexibility, muscular strength and muscular power.
Aerobic power, by definition, is the ability to move oxygen through the body. An example of good aerobic conditioning is being able to perform series of katas continuously. This is probably the most common attribute many people work on.
Anaerobic power basically is the opposite of aerobic power: the ability to work without oxygen. Delivering quick punching and kicking combinations is anaerobic power at play. The importance of anaerobic conditioning is to be able to efficiently perform rapid movements and techniques (such as combinations) over a longer period of time.
Flexibility mainly refers to the range of motion of a particular muscle, muscle group and/or joint. Obviously, some techniques require a larger range of motion than others.
Muscular strength and muscular power are commonly confused with each other, and also sometimes incorrectly used interchangeably. Strength is basically the amount of force that can be generated in one all-out movement; power is how quickly that force can be generated in the shortest amount of time possible. How much weight one can lift would indicate strength; how quickly they can lift said weight would indicate power.
Putting it all together, the best way to excel at a particular martial art would be to maximize the more important corresponding physical attributes needed for that style. I've compiled a list of styles with some of the important attributes needed. While this isn't by any means a 100% complete list, it does give a general idea.
Grappling arts (aikido, jiujitsu, judo, etc.)-low aerobic and anaerobic levels, moderate flexibility (necessary for many joint locks and holds), moderate muscular strength and power (since many throws are performed due to leverage or because an opponent is trying to alleviate the pain caused by a joint lock); judo requires high levels of strength and power due to their style of throws; moderate to high aerobic and anaerobic levels are also need for judo
Striking arts (taekwondo, karate, kung fu, etc.)-moderate to high aerobic and anaerobic levels (for kata performance and sport-style sparring), moderate to high flexibility (necessary for kicks), high muscular power (a.k.a. speed and quickness) and moderate strength