Nothing I have ever learned about "proper grammar" requires the use of "on" with "cut down". But in AmE the use of "on" is far more common than its omission. Nevertheless it can sometimes be omitted in natural usage; e.g. "The only true way to lose weight is to cut down your caloric intake, increase your exercise level, or both." But I emphasize that in my example, most speakers of AmE would say "cut down on."
Student or Learner