Short of Mexico deploying their military to the border, which is as impractical as us doing it, no you are not going to stop people from doing it. Even if you did that, people would figure out a way around it. Which is why the whole "seal the border" makes no sense. I don't think people realize what doing something like that would actually do, and it would go very very badly for a lot of people.
Ostensibly the point of border patrol is to make sure people who don't have a visa or anything don't come into the country. Which is fine if you have say a hostile neighbor. Instead we're just villainizing desperate poor people who are mostly looking to avoid living in extreme poverty. So we are wasting money and resources to protect us from people who want to come here to do menial jobs?
And I'm sure you'll say that drugs and trouble come across. I'm sure that along people willing to risk their life to come here and wash dishes at Olive Garden, some have probably less than sterling criminal histories. Drugs on the other hand, the huge majority come through legal ports of entry. So even if you could stop a few enterprising people who think lugging a 50 pound pack of drugs across an unforgiving desert is a good idea, you wouldn't make a dent in drugs coming into this country. The days of guys like Sal Maglua, Jon Roberts, Willy Falcon, Mickey Munday, etc are long over.