After shutting off the power to the fixture and removing the fluorescent bulbs, remove the ballast covering, and then test the ballast using a volt ohmmeter, which should show a continuous electrical current flowing through the device. If there is electrical current, look for other signs of damage to the fixture. If there is no current, replace the ballast.
Shut off the power to the fixture by accessing the proper switch in the building's circuit breaker box. DoItYourself.com recommends waiting a few minutes afterward to ensure any lingering current has dissipated from the electrical system.
Depending on the type of fixture, the ballast cover can be removed by squeezing the cover or by removing hardware attaching the cover to the base.
After accessing the ballast, make sure there is no oil leaking from the area, which could indicate a ruptured seal inside the device. If oil is present, replace the ballast.
Test the wires coming from the ballast by touching one probe of the ohmmeter to the neutral wire -- traditionally the white wire -- and the other probe to the colored wire or wires. The meter's needle should register a continuous electrical current. If not, the ballast likely is faulty and needs to be replaced.
One way to diagnose a failing ballast is to install new bulbs in the fixture. If the new bulbs do not work, the problem probably is the ballast.