Haha, someone should go make an article out of this since I'm lazy, but here goes:

How to remove password from a level in 8 simple steps

1. Open JCS
2. Load the passworded level
3. Enter the password
4. Go to Tools --> Level Password
5. Enter the old password in the "Old password" textbox
6. Enter oesmonk in the "New password" and "Confirm new password" textboxes
7. Click the OK button
8. The password has successfully been removed from the level

How to check that the password has been removed?

1. Go to Tools --> Level password. The "Old password" textbox should have disappeared. Or,
2. Save the level and reload it. It will not prompt for the password.