It's vital and walls do work, you live in a house with locks, Mexico has walls too you know. It's not so much that we fear its a contributing factor to crime, but we have laws and everyone present in America neeeeeeeeeeds to be accounted for because their ARE murderers that cross over with a clean slate and they DO commit murders, imagine taking down your front door and allowing complete strangers in, we cannot trust everyone, what if they came in and shot you dead and took all over your belongings? You lock your doors because people in general cannot be so easily trusted... at one time no one needed to lock their doors but criminal activity has sky rocketed since.
It has always been vital to document migrants, when our ancestors migrated to America they had to go through a process, they had to gain identity and they also had to pledge to defend the nation in times of war.
The trouble is that the issue of a border wall isn't as simple as a comparison to houses and the like. Some have argued that if walls don't work it must mean prisoners stay in jail voluntarily. It's clearly absurd to think that if you took away the walls of a prison all the inmates would stay out of a sense of duty to complete their sentences, but it isn't a like-for-like comparison. A prisoner doesn't have access to digging equipment, nor are there areas of the prison wall where a prisoner can spend hours at a time digging without being noticed by a guard. Unless you're going to post guards along the entire length of a border wall the chances are somebody is going to dig underneath it, especially if there is a tangible benefit in doing so. We've already seen drug runners bringing their wares into the country through tunnels so it shouldn't be a surprise that it's possible.
Prisoners also don't have access to aircraft and handy parts of the prison boundary that aren't patrolled very often. The border wall can be bypassed by anyone with access to an aircraft - they just hop right on over.
Prisoners also very seldom get chance to leave the prison. When a lot of illegal immigration is a result of overstaying a visa it's just another way the comparison isn't valid. If a maximum security prisoner could check himself out of prison to go shopping for the afternoon, the comparison would work better.
It is important to make sure people who aren't supposed to be here don't get here and are removed if they are found. The trouble is that a wall is only one part of the process and that criminals tend to go for the lowest hanging fruit. For many would-be illegal immigrants the chances are walking across the border, through the desert, hoping they survive the journey, probably isn't the most obvious method of choice to cross the border. It seems easier to hide in the back of a truck that's headed north - that way even if you get caught you're dropped off near the border ready to have another go.