I think that Arnold mentions Sophocles because he was an ancient playwright who understood the tragedies of the characters in his plays much like the speaker understands the tragedy of the loss of religious faith. Both Sophocles and the speaker are deep thinkers who can all too clearly see the problems around them and seek solace from them in either love or art.
In general the poem is a very Victorian one which is concerned with the withdrawing of religious faith in Britain with the advent of modern science. Britain is being compared to Greece with the speaker imagining Sophocles gazing out to sea thinking, much as he is himself.
Dover Beach is a dramatic monologue with the lover as the speaker and the ladylove as the silent interlocutor. The poem aesthetisizes the apprehension of a sensitive spiritualist at the erosion of faith from the Victorian life which had become crassly materialistic. It is also an elegy ,for the speaker-poet mourns for the recession of faith in Christianity.The last two lines express the slow rhythm of the conflict that cause sadness. He uses allusion "Sophocles long ago...."in the starting of the poem by refereeing to his story that said that atheism would cause misery. - See more at: http://www.chacha.com/question/what-does-the-speaker-have-in-common-with-sophocles-in-matthew-arnold's-poem-dover-beach#sthash.1rQywPhX.dpuf
They both hear in the sea the sounds of eternity and pervasive sadness