I suppose if you play a lot of badminton, your overall fitness will go up. But I think the better way to put it would be: Long distance track helps badminton. Afterall, to improve badminton/track, you'll need to do a lot of physical exercises. You're talking about the game of badminton alone, or the physical training that badminton players do outside the court?
It may help your lungs but not your legs (much). Middle to long distance running utilizes slow twitch muscles. Badminton utilizes fast twitch muscles.
Long distance track might improve your overall fitness, but it wont have any effect on your badminton. To improve your endurance for badminton, interval training is much more effective. A good exercise to improve badminton endurance is as follows - sprint 20-35 metres - jog back to starting point - repeat 10 times immediately - rest for 5 minutes - decrease distance slightly and repeat exercise - rest - decrease distance again and repeat Even better is to do the exercise on a hill, sprinting up and jogging back down. You should feel pretty shattered after doing this.
it only hurt your training if u wanto be a body builder. cardio + strength not always go hand-in-hand for that...