No, we cannot have absolute zero illegal immigration. Thats too obvious for you to really be asking seriously, but what the hell I'll play along with your game.
We must be able to control, AS BEST WE CAN, who enters our country. Building adequate systems to prevent illegal crossing of our borders, enforcing existing laws, including prosecuting companies who hire illegal immigrants and abuse them for their own financial gain, and lastly getting more than lip service from those whose responsibility it is to do so.
Did you watch the video posted? Have you read any statistics?
Care to elaborate on that?
Oh and a guesstimate on total aid given to Mexico is probably somewhere around $60 million a year.