If you have a voltmeter measure the voltage at the battery and then at the solenoid. Should very close to the same. Any significant voltage drop, start looking for loose, worn or corroded cables. There should be a smaller wire attached to the solenoid on the "S" lead. Take that off and have someone try turning the key to the start position. You should have a little over 12 volts on that wire and no volts when the key is not in the start position.
It will probably be less than the battery voltage but no more than a couple of tenths of a volt. Any more than that look for worn, loose or corroded cables between the solenoid up to the start switch and back to the solenoid again.
If it checks out with good voltage back to the solenoid, the solenoid should be replaced.