Apostrophe is a central device in the second half of "Frost at Midnight." Beginning in the third stanza, the speaker addresses his sleeping infant directly. Although the baby is sleeping, and so doesn't hear or respond to this address, this apostrophe nonetheless conveys the speaker's deep love and hope for his child. (Note that while the entire third to fourth stanzas are apostrophe, we've specifically highlighted the moments where the speaker explicitly begins and repeats this address—"Dear babe," etc.)
The introduction of apostrophe in the third stanza represents a significant shift in the poem. In the first two stanzas, the speaker isn't explicitly talking to anyone. He simply begins by describing the night: "The Frost performs its secret ministry, / Unhelped by any wind." As with so many lyric poems, the speaker is basically talking to the reader. In fact, the speaker talks as if the reader is right there with him, as when he describes the cry of a baby owl: "The owlet's cry / Came loud—and hark, again! loud as before." In effect, the speaker asks the reader to listen as the owl cries out again. The speaker acts as if the reader can hear the owl too, as if the reader is sitting next to him in the cottage. The first half of the poem, then, becomes like one part of late night conversation between speaker and reader—one reason that this poem is typically classified as one of Coleridge's so-called conversation poems.
This changes in the second half of the poem. At the beginning of the third stanza, the speaker directly address his child: "Dear Babe, that sleepest cradled by my side." And again a few lines later: "My babe so beautiful!" And again "But thou, my babe!" Throughout the rest of the poem, the speaker talks specifically to his child, telling the baby about his life, his views on religion and education, and his hopes for the child. The poem shifts gear from the speaker's reflections on his own life and feelings of isolation (such as sitting up at night while everyone is asleep, or being homesick at boarding school as a child). Now, the speaker devotes his attention to another life, the life of his child.
Of course, the sleeping infant doesn't hear the speaker. As a result, this apostrophe isn't meant to literally communicate with the child. Rather it represents a change in the speaker's mindset as he opens himself up to hopes about nature and spirituality. It functions almost as a kind of prayer or blessing, in which the speaker articulates certain goals for his child—both so that the speaker will be better able to achieve them, and perhaps hoping that God will hear and make them come true.