I think that Sophia Auld is relevant because she shows how corrosive of an influence slavery actually was. Douglass constructs Sophia's character as one who is benevolent, at first. She shows mercy to Douglass. She teaches Douglass how to read, helping to be one of the first forces that enables him to understand how to rise against the institution of slavery. She represents the one hope of redemption amongst White people of the time period. Yet, Douglass' aim in the work is to show how slavery and its institution is a form of poison. It infects both its victims as well as its aggressors. Subtly and effectively, Douglass shows how Sophia is a victim to this paradigm. Slavery and the fear it causes in the owners causes her to change, losing sight of her initial benevolence and grace. Rather, she is reprimanded by her husband out of fear that if one gives too much to slaves, one will regret it. In this, Douglass shows slavery to be an institution that is not rooted in strength, but rather in fundamental weakness. In Sophia Auld, this idea is demonstrated, making her character highly relevant to the thematic effectiveness of Douglass' work and narrative.
Sophia Auld is important in Frederick Douglass's narrative in a few ways. First, she begins to teach Douglass to read, and her instruction, though stopped by her husband, starts Douglass down the path to literacy. Learning to read is important to Douglass because it helps him understand the intellectual arguments against slavery (though he already understands how hideous slavery in a personal way), and literacy also helps Douglass by providing him with skills he can use when he is a freedman. Therefore, he has more confidence and greater will to escape to the north. When Sophia Auld's husband becomes enraged upon hearing that she is teaching Douglass to read, Douglass understands the power of reading and pursues learning by bribing local children to teach him his letters and through other means.
In addition, the character of Sophia Auld is symbolic of the way in which slavery degrades slave owners. Douglas paints her as sympathetic at first and then shows the ways in which slavery makes her into an unsympathetic and immoral woman. One of Douglass's purposes in writing his narrative was to show Northern whites the ways in which slavery degraded white slave owners, and he does this through his portrait of Sophia Auld and her trajectory from being a sympathetic white woman to becoming an immoral slave owner.