A physical barrier is necessary, but not sufficient.
You have to remove the incentives for illegal migrants.
All remittances should be confiscated to pay for the cost of illegal migrant remediation.
All employers of illegal migrants should be fined to pay for illegal migrant remediation.
Illegal migrants should not receive any government subsidy.
All illegal migrants should be fingerprinted, DNA’ed, and facially recognized. Tattoos would be nice — maybe florescent?
Yes. Everything you said.