Juan lives in Mexico and works in a perfectly competitive market. Whether Juan works in Iowa or Mexico he can produce the same marginal product. If the wages are the same in both places, will it be rational for Juan to work in Iowa if the dollar is depreciating?
I think Juan would be best to stay in Mexico if Juan is sending money home to support his family. If the USD is depreciating, then the USD is buying less goods in Mexico. Juan would be better off staying in Mexico where his marginal product equals his wage. Otherwise, Juan will be working longer to buy the same amount.
If the dollar continues to depreciate, then border patrols will not be necessary. Only those immigrants who want to live in the US will obtain the necessary paperwork since they want to live here for reasons other than pecuniary.