Use your stiff brush to clean out the joints around the trim, where it meets the wall on the outside and the door casing on the inside. Remove all loose debris and dirt.
Put the caulk tube in a caulk gun. Use your utility knife to cut 1/4 inch off the end of the tube. Make the cut at a 45-degree angle.
Stick a long nail into the hole so it breaks the seal inside and releases the caulk. Compress the caulk gun trigger until caulk comes out of the end. Hit the release latch at the back of the gun so the caulk stops flowing.
Hold the tip of the caulk tube against one end of the sides of the trim, on the outside. Hold the gun back at a diagonal angle. Compress the trigger so the caulk emerges from the tip.
Drag the gun slowly back along the edge of the trim where it meets the wall, letting the caulk fill that space. Lay the bead for the whole length of the trim.
Lift the gun off the trim at the end of it, and press the release latch to stop the caulk flow.
Wet your thumb and run it gently over the length of the caulk line.
Repeat the process for each outer edge of the trim, and each inner edge, where it meets the door casing.