Instructions
Start up your PSP as you normally would by pressing the power button and waiting for your home screen to load. Then scroll over to the "Tools" section of your cross-media bar (on the left side) and then select it with the X button.
Scroll down the tool menu until you see the "Settings" tab. Select it, and then scroll through the various settings with the up/down buttons.
Find the Javascript settings from the list mentioned above. Use the X button to highlight the "Enable Javascript" setting and then highlight the word "yes" using the D-Pad. Press X again to confirm your decision, and then press the O button to exit the settings menu. Scroll over to the browser again using your directional pad. You should be able to view Java on most websites now.