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:

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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.