To repair a baseboard heater's thermostat, use a multimeter, continuity tester and screwdriver. Make sure that the power in the home is turned off and all breakers are down before beginning.
When the power is off, remove the cover of the thermostat and lift it away from the junction box. Once the wires are exposed, use the multimeter to ensure no power is moving through the wires. As you disconnect the wires, be sure to label them as going to the load or the line; once they are all disconnected, check them with a continuity tester. Move the dial from low to high and pay attention to the light. If the light does not illuminate, a new thermostat is needed.
To install the new thermostat, attach the load and line wires to the corresponding areas on the thermostat. Tuck the wires into the box, reattach the thermostat to the junction box and finish by reattaching it to the baseboard.
Use caution during the process. When attempting this repair, be sure to use a high-voltage thermostat designed for baseboard heat. These thermostats use the same voltage as the heater, typically 240 volts. A low-voltage thermostat does not work and can be dangerous.