Upvote:4
A lawman's memoir describes the U.S. erecting in 1959 chainlink fences at canal crossings in Mexicali to prevent rabid dogs entering. There were plenty of cases of rabies in Texas at the time. Rabies is a real threat, though fears of its entering the country may have been overblown or even racialized.
California may have asked for a fence, but it appears to me that Congress never took up the issue. Wikipedia suggests that the first border barriers were built in 1994.
It may be worth noting that the WHO's 1987 Guidelines for Dog Rabies Control offers nine "principal" techniques for preventing transmission across borders, none of which is a border fence.