Mexican influence on American society,. i.e. illegal immigration, or American influence on Mexican society, i.e. demand for drugs?
I would go with a third option: Mexican influence on Mexican society.
I mean, damn, those poor people trapped in a country filled with outrageous government corruption from top officials to the cop on the street, violence, massacres, kidnappings, murders, touching every corner of the country, and every citizen. It is a Catch 22 situation. When they try to get away, they become "America's problem"; when they stay, they become victims.
I fondly recall my visits to Mexico many years ago in my youth as delightful escapades meeting friendly, smiling people who had the best danged food on earth. Last time I was there, about 30 years ago, it was a dank, dirty, graffiti-ridden hole, where policia so routinely targeted vehicles with USA plates for shakedown that we later learned (after having a car of thugs ram our huge RV, and a bunch of goons surrounding us demanding dinero on the spot) that if one must cross the border, park all vehicles and walk across the border, leaving anything of value behind.
However, we never went back again. Such a shame. Mexicans are among my favorite people for their vibrant culture, cuisine, music, and (once) simple joy of living. It's mostly gone for those people now, because they were betrayed by those charged with protecting them and their way of life.
/rant
Sorry. The relentless destruction of that country from within during my lifetime really, really makes me sad... and madder than hell.