Measure the width of the handle hole in the storm door, using a tape measure.
Apply a bead of exterior-grade caulk around the rim of a button plug, using the hole's measurement to choose the correct size button plug from a button plug kit. A button plug kit contains many different size button plugs.
Push the button plug with the bead of exterior-grade caulk into the handle hole on the exterior of the storm door. Clean the excess caulking from the door and plug's surface with a rag.
Fill the hole in the storm door's core with nonexpanding foam, using the handle hole in the interior side of the door for access. The foam insulates the door and prevents condensation. Nonexpanding foam will not push the button plugs out of the storm door as it dries.
Run a bead of the exterior-grade caulk around a button plug. Push the button plug into the hole in the interior side of the storm door. Clean the excess caulking from the door and button plug with the rag.