Government funding does make it a government action, however, it's important to note that the law in the story that funded the sentinels could have resulted in unintended consequences for the bad future.
Considering that the Sentinels were capable of detecting mutants at the time and had been since their introduction 16 years prior to the publication of this story. It would make sense that in the wake of an assassination of a U.S. Senator by known mutant criminals, the general public mood turned to favor a screening system that would want better screening and protection against mutants... similar to how the President riding in convertible Motorcades stopped following Kennedy's assassination in real life OR the increase of TSA restrictions on items one can take on airplanes following the 9/11 attacks.
Thus the Sentinels were funded to act only as a security deterrence. At the time of the story, the Sentinels were the only system that could identify Mutants AND carry enough onboard weapons systems to neutralize them. It would have been sold to the public as a security tool. Nothing more than a cop with a gun and a metal detector.
Of course, the Sentinels being an AI, had in prior stories gone off mission from taking their job to an extreme (Their debut story had them turn on non-mutant humans because non-mutants gave birth to mutants and thus were the source of mutants... they were only stopped when Cyclops tricked them by exploiting this logic and explaining that humans weren't the true source of mutation, the sun was, which convinced the entire fleet of Sentinels to try and stop the sun... and went about as well as you would think). This is a consistent fatal flaw, and by the 2013 storyline, it's implied Sentinels had gone off the rails, usurped the U.S. government, and were now working to nuke the world because killing everyone would stop mutation from occurring. Essentially, it was a Skynet scenario where the computer AI was operating on a "Garbage In, Garbage Out" programing, and by the time the people in power could do anything about it, the Sentinels had too much power to be stopped.
At one point in the storyline, future!Kitty walks by a graveyard and narrates that by this point, the Sentinels had targeted the non-mutant heroes of Marvel as well (including Spider-Man, Dare Devil, the Fantastic Four, and a good number of the Avengers (at the time of publishing, none of the heroes could be counted as mutants because Mutants were defined in the Marvel comics as people who were innately born with their powers, whereas all of the heroes were non-mutants who developed their powers due to outside forces they interacted with.).
As such, it's possible the law was compliant with the Constitution (After all, you do have to go through a metal detector when you go to Congress and other government buildings... why not a Mutant detector... you know... just to be sure...).