I don't think it will do one thing to illegal immigration. Zero, nada, zilch. Illegal immigration isn't something Mexico can fix in the short term. It is doubtful they can fix it in the long term. They can't even stop drug cartels and corruption. They're probably making more money from the flow of narcotics than they are from the flow of legitimate goods.
The whole "Mexico will pay for the wall" comment has always been misrepresented and overblown. There isn't a bill that can be presented, an account debited. The concept of Mexico paying for the wall has always been in overall GDP impact - both here and there. It is exceptionally difficult, if not impossible, to judge who would actually foot the bill in any real world implementation of such policy. It is easy to determine who will pay for any wall directly - we will. In the form of more issued debt.
As I have mentioned before in regards to China, tariffs are flawed because of the simple fact that only one man in political power is behind them - Trump. Therefore, China and Mexico and whatever companies affected need only wait for this one man to be out of the picture before business can return to normal. More and more this looks to be sooner rather than later as an embattled Trump continues to marginalize and isolate himself with these dramatic moves. Additionally, all interested parties now have an interest to get rid of Trump rather than support him or remain on the sidelines. And there are some pretty powerful players in there.
If we were united, Congress and the White House, in imposing tariffs that would be different. The message would be "this isn't going away and you need to change something or we're all going to be at this for a long, long time." But the Democrats and many Republicans are set against Trump no matter what he does - even if what he does is right and needs to be done. This is the whole problem with our politics: Team before Principle.
All Mexico has to do is close its borders to the migrants. Put Federal police and harden it with lots of barriers. Andres Lopez-Obrador is full on Socialist so, he is deliberately, subverting the US by letting the migrants in and busing them to the US border! No sovereign country will tolerate that crap! Mexico is being a bad neighbor to the US by creating problems which could have easily been avoided! They have tens of billions of dollars surplus with the US and to act in a hostile manner to your neighbor is totally, unacceptable! Remittances to Mexico needs to be another option if the Mexican government still will not take action! Do not bite the hands that feeds you!