It is no secret that the Cartels essentially run the nation of Mexico, and with a plethora of potential Casae Bellae that could stem anywhere from America's War on Drugs launched by the Ronald Reagan administration to the Trump administration claiming a Casus Belli of "national security" citing the ever-growing threat that the cartels have presented as they have quietly taken over the Mexican government and very firmly control the Mexican states, in fear of them setting their eyes on the US/Mexican border states as they have, but many ask the simple question of "how would this war pan out?"
The biggest ordeal out of this is the fact that the U.S has continually manned the US/Mexican border, since the Mexican-American War of the 1880s/90s and then some, with American soldier records manning the border between the two countries as far back as during Great War (WW1), the Interwar-era of the 1920s/30s and even during the Second World War and further on as the Cold War started. The situation has a potential to be as big as the entire Global War on Terror, or a strange combination of the US intervention in Afghanistan and the Vietnam War.
The How: How would the US launch these operations, and how would they look?
The US would likely begin operations via expanded aerial reconnaissance via drones such as the RQ/MQ-9 Reaper, before growing them into limited aerial combat operations to strike key Cartel logistical centers such as weapon caches, drug stockpiles or manufacturing facilities, etc; potentially expanding this primarily air-based operation to some limited ground operations like "snatch and grab" runs against High-Value Targets [commonly referred to as HVTs], limited CIA / Special Forces joint task-forces on the ground, kind of similar to that of the Vietnam-era MACV-SOG Special Operations Unit. Ignoring ground-based operations here; the U.S has aerial dominance here on a pretty realistically easy-win. If there is any resistance from the Mexican Air Force, it would come from five (5) Northrop F-5E Tiger IIs, and Thirty-Three (33) Pilatus PC-7s, and up to potentially Fifty-Seven (57) T-6 "Texan II" U.S-built Pilatus PC-9 basic trainers, which EVERY American pilot would be incredibly familiar with, given the fact that it is the base-model flight training aircraft for the U.S.A.F.
While Ninety-Five (95) light fighters and advanced trainers seem like a solid defense on paper, the majority of these aircrafts lack major Air-to-Air capabilities such as an on-board radar system, and they all lack the capability to engage and fight Beyond-Visual-Range (BVR), whereas every U.S.A.F F-15, F-16, F-22 and F-35 most certainly; without a doubt, have these capabilities, and the fixed-wing fleet of the Mexican Air Force would more than likely be grounded permanently, never being able to get up and fight; or absolutely squashed in what may be the most high intensity aerial combat seen since WW2.
But what about ground-based Air Defenses such as Surface-to-Air Sites and systems? On paper, the Mexican Government does not have anything outside of a handful of older first & second generation, American-made Stinger Man-Portable Shoulder-launched Air Defense Systems [MANPADs for short], and outside of that, the Cartels have been seen utilizing AAA [Commonly referred to as "Triple-A" or Machine-gun fire] off the backs of improvised uparmored technicals built off quarter-ton, half-ton and three-quarter ton pickup trucks. While AAA-fire, early-generation MANPADs and the occasional Rocket-Propelled Grenade (RPG) launcher helps, this would really only pose a threat to American rotary aircrafts such as the MH-60 Blackhawk helicopter, or the CH-47 Chinook
The Ground-based operation(s), be it a limited operation with Special Operations or the highly unlikely potential of a full-scale invasion that would require the "all hands on deck" method from not just the US Air Force, the Army and limited naval support, but also the added support from the Central Intelligence Agency (C.I.A), DEA and other various aides from intelligence apparatuses, while there is a potential of Mexican Army equipment falling into the hands of, or being used directly by the cartels in a conventional war setting; there is something that needs to be made aware of, and that is the fact that the Mexican cartels have, and have done for such a long time; and that is infiltrate the Armed Forces of the U.S. Both the US Marine Corps and the US Army, the Mexican cartels have utilized Uncle Sam's finest as a means to train a lump-sum of their footmen, only to have them get out of the US Armed Forces and turn around to train others within the cartels. While it is a common thought from American veterans of the Global War on Terror [or GWOT for short] that the cartels are just mirrored images of the Afghanis they thought, and while the terrain might be similar, it is far from the truth.
The cartels are not just a handful of gangs or Afghani sheepherders with a hodgepodge mix of antique war relics and Soviet-era weapons, the cartels, if this hypothetical ordeal became a reality, would be an interesting ordeal for the United States as they would be fighting both, a conventional and asymmetrical war simultaneously, and while this fight would be in America's backyard; it has the very real potential to hit a vast majority of people close-to-home in border states and further inland as cartel cells, gangs affiliated with the cartels or even cartel operatives running their own groups in states further inland, could quickly turn into "sleeper-cell" style units or pocketed resistance that could either enact terrorist-style acts in retaliation for the war against the ever-armed populace of the United States, or they could spark coordinated attacks with local allied/cartel aligned gangs to hold up smaller towns or even attack defense-related industry or manufacturing facilities.
At the end of the day, the most ideal solution is keeping the cartels "at-bay", and getting the Mexican/US border under control. While a border-wall may, unironically be the best bet, nothing beats the utilization of modern technology that the U.S most certainly has at it's disposal, such as FPV drones for short-range reconnaissance, RQ-series drones for longer range monitoring and surveillance of the border, along with a potential, mutual agreement between the U.S and Mexican governments for limited actions around the border to minimize the ever-growing effects of human trafficking and the fentanyl-crisis, or at least to put a figurative "dent" into those specific operations while also working within the mainland United States to push out cartel influences. While the aforementioned potentials of limited ground or aerial operations, or total-war scenarios are unlikely, and it is incredibly unlikely to be exactly like a modern version of the Vietnam War; it is tough to say exactly what could happen as there are simply too many variables and potentials that could/couldn't happen based on the responses and actions of those in power at the time.