This is actually US policy for anyone who goes to jail for any reason, not just people whose specific crimes are illegally crossing the border. If you go to jail, and you have a dependent child with no other care giver - the child will go into the custody of the state.
Also, keep in mind that it isn't all asylum applicants who get separated. If you show up at a valid port of entry into the US and request asylum, then no crime has been committed and no one is separated from their families. Only people who sneak across the border illegally(which is a crime) are processed from their criminal behavior.
Going to a valid port of entry also happens to be a much cheaper and safer way of getting to America, considering it doesn't require the use of human mules, paying exorbitant fees and coordinating with drug cartels, and performing dangerous desert crossings where you may get abandoned if your mule gets spooked.
edit: alasdair_, I can't comment on your post thanks to HN aggressively preventing a poster responding to numerous comments on a post, but I would imagine the reason that she couldn't file an asylum claim is right in the article: Because she lied about being a US citizen in the past, and that bars you from presenting for asylum. And finally, the father separated himself from his children before he even entered the US, because he knew he couldn't enter the US because he was already deported from the US in the past.
> Immigration and Customs Enforcement (ICE) officials say Maria made a false claim of US citizenship in 1999 and served a 75-day sentence in a federal prison before being deported back to Guatemala at that time. In a statement provided to CNN, ICE says that lying about citizenship status "results in a permanent bar from being able to legally enter the United States."