In a word, ABSOLUTELY! First my "how's", then some proof. You don't have to do massive squats or deadlifts or anything overly intimidating. However, you do have to push yourself. Strength training your entire body will help you through a run, of any distance. You need a strong upright core to insure proper and efficient running form and breathing. Without a strong core, as your fatigue increases, your form suffers and you become less efficient, and your fatigue increases more! Strength training your lower body has a number of positive effects. You will decrease your risk of injury, your recover times will be faster, and your actual running will improve. Upper body training is equally important. As you fatigue, if you don't have a strong back and shoulders to maintain proper posture, you could begin to slouch, which could prevent you from getting good air into your lungs, it could also limit your hip flexor range of motion, which would prevent efficient running technique. I would suggest single leg training. We don't have to load our backs with super heavy weights to gain muscle in our legs. Plus single leg exercises help with balance and works the smaller stabilizing muscles. Plyometrics are equally important, but I would incorporate these in the later stages of training, after you have built up a good base of strength, otherwise you increase your risk of injury.
Now, how do I know this??? Well, I have a little bit of knowledge from research and experience, and I've trained runners. Most recently I have been working with an ultra-marathoner (100 milers!!). We have been training his legs very heavy. His last run, he said a lot of elite runners were saying this is one of the hardest 100's out there; he wasn’t sore AT ALL the next day! Usually you're a walking mess the day after. Also, he only runs about twice a week! And one day of strength training with me. THAT is amazing, and none of the other runners believes that's all he runs. He said his recovery times are down (See above), and his injuries have lessened or gone away or don't return. And his hill climbing is amazing compared to where it was. His downhill speed is increasing and he feels more stable coming down as well.
Granted he is an elite runner, but strength training will help any runner get better.
More Answers from Michael Ogawa , NASM Elite Trainer