Great...come legally. If you don't, you should be turned away at the border immediately.
Actually, there is nothing illegal with approaching a border and claiming refugee status. That's how it's done, especially when fleeing for one's life, as are many if not most of those leaving Central America. And under the UN Refugee Convention, which is law in the US and most of the world, it is illegal to turn away a refugee claimant. Now, once their claim is adjudicated, if it is found to be unfounded, then they can be deported but not before then, otherwise it would be the US who would be acting illegally.
However, there is another way to approach this. Refugees are supposed to make their claim in the first safe country they enter and so if Mexico was considered a safe country, they should make their claim there, which many do. And the US could make an agreement with Mexico, as we have with the US, to return refugee claimants to that safe third country, in the terminology, while their claim is adjudicated in the US, as we return refugee claimants coming from the US and making a claim in Canada to the US to await their decision. Some of them anyways. It's a little more complicated than that.
As I know that you like to get things right, here is a link to the UN Convention so that you can educate yourself on these things: