It is important to realise that one of the central themes of this excellent novel is that of childhood and growth. Ponyboy, as the youngest member of the gang of Greasers of which he is a part, is surrounded by a number of people who have grown into bad examples. Throughout the novel we always have in the back of our heads a concern about what Ponyboy is going to grow into, and whether he will be able to escape the many factors ranged against him and make something of his life. This issue is directly related to the Frost poem:
Nature's first green is gold,
Her hardest hue to hold.
Her early leaf's a flower;
But only so an hour.
Then leaf subsides to leaf.
So Eden sank to grief,
So dawn goes down to day.
Nothing gold can stay.
Looking closely at this poem it is clear that this is all about the theme of growing up. Nature is used as an analogy for the innocence of childhood and how it is lost with time. The "first green" is actually "gold," but this is only very temporary. The final line, "Nothing gold can stay," reflects the inevitable loss of innocence that all of us face as we grow and mature.
Note how Johnny in his final letter to Ponyboy expresses his interpretation:
...he meant you're gold when you're a kid, like green. When you're a kid everything's new, dawn. It's just when you get used to everything that it's day. Like the way you dig sunsets, Pony. That's gold. Keep that way, it's a good way to be.
The poem thus concerns the struggle in all of us to retain some of our "golden hue" before "leaf subsides to leaf" and we lose our innocence and child-like wonder of the world completely. The Frost poem helps reinforce the central message of the novel, which is based around the series of choices that we have to make that will decide who we will become when we are older.