Secure the Border Act of 2023 This bill addresses issues regarding immigration and border security, including by imposing limits to asylum eligibility.For example, the billrequires the Department of Homeland Security (DHS) to resume activities to construct a wall along the U.S.-Mexico border;provides statutory authorization for Operation Stonegarden, which provides grants to law enforcement agencies for certain border security operations;prohibits DHS from processing the entry of non-U.S. nationals (aliens under federal law) arriving between ports of entry;limits asylum eligibility to non-U.S. nationals who arrive in the United States at a port of entry;authorizes the removal of a non-U.S. national to a country other than that individual's country of nationality or last lawful habitual residence, whereas currently this type of removal may only be to a country that has an agreement with the United States for such removal;expands the types of crimes that may make an individual ineligible for asylum, such as a conviction for driving while intoxicated causing another person's serious bodily injury or death;authorizes DHS to suspend the introduction of certain non-U.S. nationals at an international border if DHS determines that the suspension is necessary to achieve operational control of that border;prohibits states from imposing licensing requirements on immigration detention facilities used to detain minors;authorizes immigration officers to permit an unaccompanied alien child to withdraw their application for admission into the United States even if the child is unable to make an independent decision to withdraw the application;imposes additional penalties for overstaying a visa; andrequires DHS to create an electronic employment eligibility confirmation system modeled after the E-Verify system and requires all employers to use the system.