You'll find people of Mexican ethnicity
widely represented in ICE, Border Patrol, and other such orgs. People looking in from the outside might find this odd, but it's really quite simple. Getting into the US legally is
hard, expensive, time consuming, and oh god - the paperwork. And eventually gaining citizenship makes gaining entry look like the easiest thing ever. So you go through all of this, and then somebody else pays some smugglers (enriching some of the worst of humanity), crosses illegally and now 'Yay - Yo soy Americano, hombre!' How do you think the former is going to feel about the latter?
It's also not just some American or Mexican thing. The same is true in many expat communities (of Americans) around the world. Actually maintaining your visa and other stuff in many places is frequently a massive PITA, expensive, time consuming, and so on. If somebody's there with the claim 'No you see bro, don't you understand I'm just an "undocumented migrant"', he's going to be held in very poor regard by most people there legally.
So even at the most basic level - illegal immigration is deeply unfair to people we want in the country. And that's just one aspect among many. The tales of things gone wrong, or simply of the emotional appeal of somebody trying to make a better life for themselves, can be very appealing - but it's but one dimension of an issue that has affects many, and has many consequences.