How is the poem "Dover Beach" by Matthew Arnold link into the theme of the moon?
Well, let me start my answer by saying that I don't really think the moon is a "theme" as such in this excellent poem. Rather, I believe it is an important image that is referred to in the poem at other stages. Firstly, however, let us focus on how the moon functions at the beginning of the poem:
The sea is calm tonight.
The tide is full, the moon lies fair
Upon the straits--
Note how the moon is part of what is used to create a peaceful and tranquil mood. Later the moon makes the cliffs of England "glimmer" in its light. However, this mood is quick to change in the first stanza as we hear the tide of the sea which is capable of changing the world by its power of altering the shape of the shore:
Only, from the long line of spray
Where the sea meets the moon-blanched land,
Listen! you hear the grating roar
Of pebbles which the waves draw back, and fling...
Of course, the tides are controlled by the moon and so we can argue that this image of the tides violently "flinging" pebbles with a "grating roar" presents the moon as a strongly active and vibrant force that changes geography in its wake. The analogy that Arnold makes later on in the poem compares religion to a "Sea of Faith," and thus subject to the same violent changes as the tide. Thus we can say that the moon functions not necessarily as a theme (though some may identify it as such), but as an important image that allows Arnold to make his point concerning the changes that were happening in religion at the time of the poem.