Remove the door from its hinges with a screwdriver. It will be easier to get all the holes if the door is on a flat surface. Lay the door across a table or a pair of sawhorses.
Fill in the holes with wood putty. Since pin holes are small, you will not need much product. Take a small spatula, and spread the putty thinly over the holes. You can also use a butter knife or a razor to scrape away the excess putty. Let the putty dry according to the manufacturer's instructions.
Use a medium-grade sandpaper to smooth out the entire door. This will help level off the puttied spots with the rest of the door. Once you are done sanding, go over the door with a damp cloth to remove any leftover dust.
Match the wood putty to the rest of the wood door by either staining the entire face of the door or painting it. Go with the grain, and apply one light coat of either product. Allow it to dry completely before adding a second coat.
Seal the door. If you decided to stain the door, protect the finish with a clear coat of polyurethane. Apply one to two coats, allowing each coat to dry before proceeding.
Rehang the door on its hinges once it is completely dry.