Determine if the drip is recent or old. Sometimes after painting the house you will notice drips on the trim that were there prior to your new paint job. Older drips will be a little more difficult to remove. Fresh drips will obviously match the color of the new paint job and are quite easy to remove. Latex paint is the paint most commonly used for interior painting. It is water based and much more easily removed than oil based paint.
Scrub the woodwork with warm water mixed with a mild detergent on fresh drips. Use an abrasive sponge. If the drip is fresh it should come off fairly easily with a little elbow grease. Use a putty knife on larger drips. They take a longer time to completely dry and it is often possible to scrape off a large part of the paint with a putty knife.
Clean old drips with Goof Off. Goof Off is a chemical remover that works to remove paint drips without harming the finish underneath. Apply the Goof Off to a clean rag or abrasive pad and rub the drip with a slight amount of pressure until the paint begins to loosen. Wipe the area with warm water and a mild detergent to remove the remainder of the paint. If the paint is still stubborn, use the Goof Off again.
Scrape off really stubborn drips with a razor blade. Goof Off works on almost all drips but sometimes it seems to take forever. Razor blades provide a little bit of assistance. Hold the razor blade at a 45 degree angle and slide it along the paint applying slight pressure. Then use Goof Off on the same spot. Continue until the paint drips are completely gone.