Tom Kurz notes:
Taekwondo master Hee Il Cho, famous for his powerful and precise jumping kicks, says, “Weight lifting can help athletes in any sport, including the martial arts. The more strength and size you have, the better you will perform. If two people weigh the same, the one with more muscle can hit harder."
You should listen to Hee Il Cho. He knows what he's talking about. (Kurz gives the source as: Jeffrey, D. 1994. The Master of Devastating Kicks: Hee Il Cho's Routine for Fast, Powerful Kicks. Martial Arts Training March 1994, pp. 20–25, 62.)
What Kind of Weight Training?
Weight training only focuses on muscle size if that's what you're focusing on. Weights are a tool. You can use them to get bigger, or to develop strength or power or endurance.
It's important to distinguish the basic forms of training with weights:
- Bodybuilding - lifting for size and appearance; the goal is to look like, say, Arnold Schwarzenegger or Jay Cutler. This is the least productive form of weight training for martial arts: one is primarily trying to get bigger, using methods such as machines and isolation work that are generally sub-optimal for functional or sportive purposes.
- Powerlifting - lifting for strength; the goal is to improve one's ability to produce force. It is hard for me to conceive of a situation where, independent of other factors, strength is anything but a boon to martial arts practice. (The only possibility I've heard is that exceptionally strong individuals must make a conscious effort to rely on technique instead of physicality. This is akin to the curse of being naturally agile: cry me a river.) The basics include the slow lifts: squats, deadlifts, bench and overhead presses, pull-ups.
- Weightlifting - lifting for power; the goal is to improve one's ability to produce force quickly. The primary tools are the fast Olympic lifts: cleans, jerks, and snatches. Power is a derivative of strength, and everything I said about strength's applicability to martial arts counts triple for power. Exerting force quickly is a fundamental aspect of nearly all sports, particularly for striking and throwing techniques.
There are many programs for strength and power training that focus on developing those qualities without adding mass. (Some of them are martial-arts specific.) A certain amount of mass, however, is often a boon anyway for undernourished martial artists.
Benefits of Strength
Achieving a significant level of strength and power is one of the most straightforward ways to increase the effectiveness of your techniques. However, getting bigger for the purpose of getting bigger is not directly productive for martial arts. (Getting bigger will probably mean you'll get stronger, which would be good.)
But martial arts is about physicality combined with technique. Strength and power are essential components of the physicality necessary to execute any technique properly. Lifting weights is arguably the most efficient method for developing those qualities. People who say differently are either already athletic (either naturally or through prior training), or are inexperienced with weight training and shun the unknown.
Minimum Strength Necessary to Practice Fighting
Weak people have no business training martial arts. They are liable to get hurt, and will find themselves too weak to properly execute basic movements and techniques. Martial arts are about optimizing the use of strength. This is not the same as obviating the need for a baseline level of strength.
If you've been around a popular martial arts dojo long enough, I'm sure you can remember a new person signing up who is physically incapable of performing even the most basic techniques. Often they find themselves injured and re-injured, toughing out muscle spasms and sore joints in order to continue doing the activity they love. This is not healthy. Students should be required to achieve a basic level of physicality before joining regular class. Strength and mobility are of primary concern in this period, with conditioning a distant third since it can quickly be developed through regular class activities.
Take well the advice of Kurz (ibid):
People who can't put a barbell or a partner weighing at least as much as them on their shoulders and easily do a few squats are too weak to learn fighting techniques.
Test this hypothesis yourself: take six months to work up to a bodyweight barbell squat. (I'd add a 1.5x bodyweight deadlift, plus a dozen chin-ups and some heavy power cleans and presses.) Then, ask yourself whether it helped you hit harder, spar longer, pin people better, and keep a better grip on your opponent. If so, great. If not, go back to doing nothing, and with scant attention to the "problem" of excess strength, you will be smaller and weaker again.