Inspect the existing exterior caulking around the perimeter of the door frame for obvious cracks, peeling or gaps. Check for more subtle air leaks by lighting a candle, holding it close to the caulking, and very slowly moving it alongside the line of caulking. If the flame flickers, there is probably a draft in that area due to a leak. Double-check that area to be certain of the leak.
Pull out all compromised caulk with a caulk-stripping tool and/or a putty knife. If any part of a single bead of caulk is leaking, such as the entire line running down the left or right side of the door frame, remove the entire bead.
Clean the open crack in the door frame or along the door sill with a rag and warm water, and by scraping any clinging bits of caulk off with a putty knife. Inspect the interior of the crack for mildew or mold (always a possibility with a water leak), and if that is present, scrub this area with a soft-bristle brush and a mixture of bleach and water. Allow your cleaning work to dry before continuing.
Insert a new tube of exterior silicon caulk into a caulking gun and release the spring. Make a cut in the nozzle at a 45-degree angle, and far enough up the nozzle so the nozzle opening matches the width of the door frame crack(s) you must fill. If you have an existing tube of caulk already loaded, adjust the nozzle cut as necessary.
Apply the caulk to the crack in the door frame by holding the nozzle against the door frame at a 45-degree angle (matching the cut in the nozzle) and pulling the gun's trigger with even, steady pressure. Steadily move along the crack with the gun to fill it with a smooth, even bead of caulk.
Run the caulk-smoothing tool along the crack after the bead is finished to scrape away excess caulk and to give the surface a uniform appearance.