This manual appears to be an authentic manual published by the Mexican government. A translation of its contents can be found here (American source) and here (Mexican source). The pamphlet does appear to describe methods to illegally immigrate to the US and methods to avoid getting arrested once in the US, although it nominally discourages such action.
That being said, the manual is rather scanty on details, and mainly consists of (very) general ideas on what to avoid in order to reduce the risk of death. Therefore, it seems that the manual was chiefly intended as a method to reduce death rather than as a method to export migrants.
A copy of the manual was created by the right wing organisation CFIF, which cited the original address of the manual on the Mexican government webpage.
This page has since been taken down. The Wayback Machine has however archived their information, and it can be viewed there.
Therefore, we can conclude that the Mexican government did in fact publish such a pamphlet, at least on their website. Given such a fact, it is not a stretch to conclude that they also published the pamphlet in hard copy, although I have no better sources than the NY Times article for this claim.