Note there are TWO circuits for the high beams on the halogen-bulbed cars; one that operates at reduced power (about 60%) for DRL, and full power for high beams. Do you have DRLs when the ignition is on and the parking brake released? You can usually hear the relay click when it is energized and released as well.
I suspect if the answer is yes the problem may well be the relay, or the drive current to it. Testing the drive current is easy; remove the relay and use a voltmeter on the pins for the coil in the socket; you should see +12V when the high beams should be on, and 0V across them when they are off. The coil terminals on the relay itself should have some resistance across them -- not zero, but also not open (typically its in the range of a few dozen to a couple hundred ohms.)
If you apply 12V to the coil terminals (out of the car) the contracts should show zero ohms (short), and with 12V missing the same contacts should be open. The most-common failures on a relay are for the coil to fail open or the contacts weld on (in which case it's always on whether there is coil voltage present or not.)