While illegal immigration has occurred since the formation of the U.S., it wasn't until the 1990s that the government began to focus heavily on policing illegal immigration. U.S. Immigration and Customs Enforcement (ICE), which today is the most visible symbol of the policing of illegal immigration, did not even exist prior 2003.
Resources:
U.S. Immigration Timeline : This resource provides a detailed history of US immigration dating back to the late 18th century.
How Trump Radicalized ICE : This piece discusses the formation of ICE, government changes regarding immigration, and how ICE has become radicalized during the Trump administration.
Abolish ICE: History : This piece briefly discusses the current push to "Abolish ICE" before detailing how and why ICE was formed, and why it surrounded by controversy.
How Border-Crossing Became a Crime : This piece discusses how illegal immigration became a crime, primarily with the mission to curb Mexican immigration.