Border Crisis Prevention Act of 2021 This bill imposes additional requirements on asylum seekers, increases certain time limits related to detained aliens, and addresses other immigration-related issues. The Department of Homeland Security (DHS) must detain certain aliens, such as an alien convicted of an aggravated felony or believed to be a trafficker of controlled substances, beyond certain time limits for detaining an alien. The bill also provides for additional instances when DHS may extend the 90-day removal period (generally the window in which DHS has to remove an alien after a final order of removal). An alien choosing to appeal mandatory detention or detention as part of a removal period extension may do so only by filing for a writ of habeas corpus. DHS shall establish a process to determine whether an alien not subject to mandatory detention and who has tried to comply with a removal order should be detained or released with conditions. Furthermore, under this bill, an alien's credible fear of persecution must be more probable than not in order to be eligible for asylum. DHS may also deny asylum to an applying alien and remove the alien to a country that is not the alien's country of nationality, if the alien would be safe and would be able to apply for asylum from that country. (Currently, an asylum applicant may be removed to a third country only if that country is party to an agreement allowing for such removal.) The Department of Justice may appoint 100 additional immigration judges.