The answer was right under my nose. Right there in the WordPress editor was the relevant line of HTML code (placed there by the plugin “Toolset Types”):
<a href="http://horizon-residence.com/">[types field="horizon-residences-banner" alt="Horizon Residence condominium" title="Horizon Residences" size="full" align="left"][/types]</a>
It was there that I could change the title, and in-turn, the alt text (since the title was overriding the alt text).