Any process that modulates the frequency, rate or extent of apurinic/apyrimidinic (AP) endodeoxyribonuclease activity, the hydrolysis of ester linkages immediately 5' to an AP (also called abasic) site within a deoxyribonucleic acid molecule by creating internal breaks, generating a single-strand break with 5'-deoxyribose phosphate and 3'-hydroxyl ends.