I have two 1500 watt baseboard heaters wired to the same thermostat. They have worked fine for 20 years. Now one is working and one is not. What possible problems could there be and how do I check.

If it is a two pole thermostat one side of the of the thermostat might have gone bad.

Check the wiring. Any wire nut connections could have come loose and allowed the wire to burn.

One of the heater elements could have gone bad.

Check the power at the circuit breaker panel and then at the baseboard heater.

You did not mention weather it is a 240 volt setup or a 120 volt setup so when you test for voltage you will have to find out which one it is.