Mark the cut with a utility knife. If you are cutting the baseboard for a door casing, place the casing against the baseboard and scribe a line.
Cut through the baseboard as much as is possible starting from the bottom using a thin pull-saw. Avoid cutting into the wall. If you have a hard time starting the cut, hold a block of wood against the baseboard as a guide. Keep the cut square or slightly angled so that the casing will fit up to the baseboard without forming a gap.
Finish the cut with your utility knife.
Nail the end of the baseboard with finish nails that are long enough to penetrate the wall studs, about 1 inch. This is necessary only if the piece removed contained nails.