What is the criteria for "asylum" as it pertains to the present border crisis?
Same as it always is, a person must show that they are a person who is unable or unwilling to return to, and is unable or unwilling to avail himself or herself of the protection of their home country because of persecution or a well-founded fear of persecution on account of race, religion, nationality, membership in a particular social group, or political opinion, or in such special circumstances as the President may specify.
All people have a right to ask for asylum, regardless of how one enters the country.
Those who are fleeing Haiti because of the recent political unrest may qualify for asylum, but those fleeing from natural disaster do not. However, under law, it is their right to be afforded a chance to make their case. The Trump policies of MPP (which Biden tried to end, but was reinstated by District Judge Matthew J. Kacsmaryk, a Trump appointee in Amarillo) and Title 42 (which Biden has continued and not even considered repealing) create a very dangerous situation for those seeking asylum.
However, it is within the rights of the President and the Secretary of Homeland Security to extend Temporary Protected Status or humanitarian parole to these Haitians which would allow them entrance into the country, the ability to apply for work authorization, but would not give them permanent status - TPS is temporary, and can be renewed at the discretion of the president.