Thousands Cross the U.S. Border Seeking Asylum. Most Won’t Get It.

About 6,300 asylum requests were approved between January and March 31

EL PASO, Texas—When 42-year-old Angelica walked up to a U.S. Customs and Border Protection inspection booth here Wednesday, she started what is likely to be a lengthy, long shot bid to win refuge in the U.S.

Angelica and her 3-year-old granddaughter, Ruth Sofia, said they fled their home in Michoacán, along Mexico’s Pacific coast, more than a month ago amid increasing violence and threats to their safety.

As...