Addressing the question in the Title of the OP:
If properly constructed and monitored, a "wall" would accomplish quite a lot. I say "wall" because some stretches of our Southern Border would not allow an actual wall to be constructed. At best, some type of security fence, and, possibly, not even that, but electric monitoring would be required.
One example of this would be the stretch of Border that runs through the Big Bend National Park. You can search it and look for pictures to better understand. As well, in my opinion, some adjustments in a "wall" must be made to accommodate the wild life that migrates from the U.S. to Mexico and vise versa. Big Horn Sheep and Black Bears and other wildlife as well. To prevent their migrations might well lead to their extinction, and I do not think anyone would want that. It would be easy enough to accommodate these creatures and still maintain a level of Security.
As for what it would accomplish:
1) Illegal entry into our Nation by peoples we do not know, nor do we know their intentions or what health issues (diseases) they may have and would pass on to our citizens.
2) Illegal drug smuggling.
3) Illegal weapons smuggling.
4) Illegal slave trade, which is an actual problem along our Southern Border.
5) Other illegal trades such as the transportation of stolen property from the U.S. to Mexico, such as Automobiles, Electronics and other items.
6) Enhance our National Security by establishing one more barrier terrorists would have to overcome to enter the U.S.
7) Saving lives. The last report I saw said that over 200 people died in our Southern deserts trying to reach a City that was too far from their point of entry for them to walk.
8) Environmental habitat damage would be reduced. There is a huge amount of trash scattered across our Southern lands left there by the illegals and drug smugglers.
These are some I can think of with little effort.
Building the "wall" has many, many more advantages other than stopping illegals from crossing to "pick fruit" or some such. Proponents of such a wall have failed miserably to properly argue the necessity for such a wall.