Crinán was married to Bethóc, daughter of Máel Coluim mac Cináeda (Malcolm II) (King of Scots, who reigned from 1005 to 1034). As Máel Coluim had no surviving son, the strongest hereditary claim to the Scottish throne descended through Bethóc. Crinán and Bethóc's eldest son, Donnchad (Duncan I), who reigned from 1034 to 1040.
Crinán's second son, Maldred of Allerdale, held the title of Lord of Cumbria. It is said that from him, the Earls of Dunbar, for example Patrick Dunbar, 9th Earl of Dunbar, descend in unbroken male line.