Spain and later Mexico did share a somewhat murky border with the French and later the US before the Mexican War; Spain took incursions into it's territory seriously, they captured wanderers, many from the US and took them to Mexico City for trial. In places even today the border is not very defined- that is why we need a major fence.
There was no defined border before the Mexican War and yes the boundary disputes started in Texas, and in what is now New Mexico on the Rio Grande. Most of the boundary skirmishes started when Texas was preparing to become a US State, so the US was involved. Texas had already had it's battles with Mexico, the disputes at the time of Texas statehood were between the US and Mexico.
The most serious and famous raid was done by Pancho Villa, it was not the only one by far. Border raids and skirmishes did happen pretty often. There was problems with the border even then. The soldiers were mostly after Indians, but the military was also assigned to protect the border, and did chase Mexicans too. Many of the border skirmishes between Mexico and the US of the time were solved by the Gadsden Purchase, because a lot of that area was in dispute still even after the Mexican War. After the treaty of Guadalupe-Hidalgo and leading up to the Gadsden Purchase, skirmishes on the border had become so serious, many thought we were headed for a second war with Mexico. We actually bought the land to build a railroad, but that purchase did settle a lot of the problems of the border at the time. The US and Mexico have a long history of not being that neighborly, and from time to time there have been flare-ups.
Border enforcement started with retired Texas Rangers and Arizona LE and US Marshalls. The BP as we know it started over 70 years ago. It was considered serious enough that the Border Patrol was formed as a seperate agency.
You can nitpick all you want, this border has been protected in the past, even by use of the military. The border may or may not be seen as that serious by today's politicians, but it has not always been that way.
BTW, have you ever lived on the Mexican border?
American adventurers and semi-official expeditions into Spanish territory were always a problem for them and they were understandably concerned.
The activity of the US Army in controlling illegals has never been much. Indians, Mexican bandits etc. were almost its entire concern.
I have never lived along the border but am very familiar with Illegals since I have lived in Chicago for 40+ years. We have LOTS of Illegals here. More than almost any other city.