Gold prizes are earned by meeting a certain point threshold on each night (stage). You must have a high enough cumulative score when you finish the stage, so it's important to do well in each area and not spend any coins on items between areas. In general, maintaining your climbing combo is critical to getting a high score; collecting piles of coins is also necessary, but on stages where there is more leeway some of the coins can be skipped.
The first stage is obviously the easiest to earn a Gold Prize on, as it is the simplest and there is only one area. You can change your difficulty and retry the stage from the Stray Sheep bar on any night.
Finally, if you prefer to follow a video walkthrough, videos for all nightmare stages can be found at http://catherinethegame.wikia.com/wiki/Walkthrough:Overview