When the computer starts up do you see the HDD LED illuminate and the fans start? If so, the power supply is outputting 5 volts and 12 volts. The -12 volt rail may be fried but doubtful. Does the tally light on the monitor change from red or yellow to green? If not, the probable culprit is the video card.
The LED display on the monitor is triggered by the monitor sensing the Horizontal and Vertical sync signals from the video card. If either sync signal is not present, the monitor will not turn on so it display a static horizontal or vertical scan line. On older CRT monitors this can damage the phosphor layer of the CRT.
If you have another computer, try connect the monitor in question along with the same cable to check for operation.