Halting immigration would help thwart some diseases, but I don’t expect the mosquitoes to cross borders through a customs area.
Zika is feared to cause Guillain–Barré syndrome in adults and microcephaly in newborns. Hopefully, both fears prove false. A concern is that vaccines for expecting mothers will be difficult to develop (risk of vaccine causing damage to fetus; need for trials on pregnant women nevertheless).
Zika is obtained by humans from mosquitoes, not from infected humans.