I expect the reason for the falling birth rate is that so many of Mexico's young people are living and having babies in the US. Of course that's going to cut down on the birth rate. But all those babies are being had in the US. So I expect the number of Mexicans crossing the border may decrease, but it will just be replaced with Mexican babies being born in the US.
You would be right. Vdare (banned from FR) published a well researched piece on this a few months ago and just a few days ago on FR we were discussing a thread about how a ridiculously high percentage of US births were to Hispanic immigrants. I don't think the thread identified them by illegal status but the majority of them are illegals.
Unfortunately, Hispanics as a group do a poor job of educating their children. Here is a disturbing study which shows that even by the third generation, children of Hispanics are 3 times more likely to be high school dropouts as children of immigrants from other racial groups/ethnicities/cultures (Table 5 on Page 10). It does not bode well for our future. Check my tagline!