Border Security and Enforcement Act of 2023 This bill addresses issues regarding immigration and border security, including by imposing limits to asylum eligibility. For example, the bill limits asylum eligibility to non-U.S. nationals ( aliens under federal law) who arrive in the United States at a port of entry; allows for 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; requires a fee of at least $50 for each asylum application, except for certain applications involving an unaccompanied alien child; authorizes the Department of Homeland Security (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; and requires DHS to create an electronic employment eligibility confirmation system modeled after the E-Verify system and requires all employers to use the system.