If the US were to annex Mexico, we would be far worse off. Incorporating third world countries into the US would cost us more than it is worth.
Which is the dying & weak civilization? The US or Mexico? I would posit that with US remittances constituting their major source of capital, it is Mexico that is weak.
If a significant percentage of Mexico's people are residing in the US, with no net benefit to us, how would taking over one of the richest regions on earth, coupled with an incredibly low population density (outside of Mexico City - read: empty), be a net cost?
It wouldn't. It would usher in a tremendous population flow slow from the old, high tax states to the 15,000 miles of virgin Mexico coastline.
In a way, this transformation is already underway. When you drive south of TJ in Baja, you can only tell where you're at by the infrastructure; the mix of anglos & hispanics is no different that when you cross back north into San Diego.
It wouldn't solve anything, anyway. All it would do is create a 'new' border that would need constant watching. Mexico watches its southern border because 'gasp' they don't like people crossing it illegaly!
Mexico, on the other hand, is not so generous when its national sovereignty is being violated. Since 1974, anyone found entering Mexico illegally is subject to two years of incarceration, if they are lucky. For years Mexican troops on Mexico’s southern border have used land mines and assault rifles to deter Central American immigrants from entering Mexico. Anyone trying to cross Mexico’s southern border illegally can be killed with little or no notice to the media.