if i was in the position where i needed asylum, i would take the time to find out what procedure my intended new country would want me to follow. i think following a nation's laws is important, especially if i'm trying to escape to that as my new home.
i don't know all the specifics of what's happening in mexico and elsewhere, but it seems to me that sneaking over the border (instead of following whatever procedure and laws may be in place) is going to carry a good deal of risk with it. is it going to be worth it to me? i'd better know ahead of time and be prepared to deal with those consequences, whatever they may be.
i also realize not all people may have the time or resources at the time they decide to run to follow protocol. they, too, had better be prepared for whatever consequences may follow.
i was never really aware of US policy for this issue until the last year (and I'm guessing most here weren't well versed in it, either, until it became a national story). if there's a way to reform the process on both sides, it's worth doing.