That link for a previous answer that Marc H. posted in the above response describes a situation where the ice chute door is not closing all of the way. You indicated that the door closes all of the way. I suspect that you are referring to that ice chute door. If that door is closing properly, I recommend that you check the seal of that chute door. Shine a light from the outside of the door and look down that ice chute to see if you detect any gaps. If you do, then that ice door will probably need to be repaired or replaced. If you have the same type of dispenser that is described in that previous answer then that information may help you complete that repair.
Be sure that you
unplug the refrigerator
before accessing internal components.
You can order parts from this page:
Kenmore Refrigerator Parts
If you find no problems with that ice chute door, then I recommend that you examine the freezer door gasket. You could have a loose spot or damage to that freezer door gasket that would allow the warm moist air to leak into the cabinet and cause that frost.
I hope that this information helps. If you need more assistance, reply with additional details and the full model number of your Kenmore refrigerator.
If you get to the point where you need to have a service technician diagnose and repair this failure, you can schedule service through this link:
Sears Home Services