Out of the vast array of Star Trek TV shows and movies, for a lot of people the two highest rated shows are The Next Generation and Deep Space 9. There are of course a plethora of die hard fans who adore the Original Series, and a smattering of people who favor Voyager, but the intrepid adventures of Captain Picard and Commander Sisko hold a special place in the hearts of many fans. What people seemed to love about the two was that they captured the two different sides of Star Trek. TNG was the utopian pinnacle of humanity, following on from The Original Series’s mission to sneak out new like and explore the galaxy; meanwhile, DS9 focused on the political underbelly of the Federation, as well as the grim reality of war for those fighting on the front lines.
The two shows crossed over from time to time, with the memorable meeting of the two captains in the pilot episode of DS9, but other than that, they were pretty separate. This raises a question, however. In the destructive war between the Dominion and the Federation, millions of Federation ships were destroyed, and the lot of them were nearly wiped out. So where was Picard and the Enterprise E during this time?
The Dominion war is one of the most catastrophic conflicts in Federation history, and is treated to be an end to the once utopian Federation altogether. During the war itself, there was plenty of reference to other ships fighting the good fight, throwing themselves into often suicidal missions to hold back the inevitable forces of the Dominion. The main DS9 spaceship, the USS Defiant, was used a lot, but there is no mention of what the Enterprise E is doing. When built, Picard's iteration of the Enterprise namesake was at the forefront of Federation technology. It was primarily a scientific exploration vessel, but also packed a serious amount of high-tech firepower — in fact, it possessed some of the most powerful weaponry Starfleet had to offer. It was, and continued to be even during the time when DS9 was set, the flagship of Starfleet. With their firepower, advanced shield technology, and proven great leadership, they should have played a massive part in the war, so why didn’t they?
As far as non-universe reasons go, it mainly came down to money. While Star Trek at this point had a fairly decent budget, bringing TNG into the series would have far exceeded their limit. While it might seem that the biggest cost in all this was bringing the crew back together, and paying them all in addition to the main DS9 crew, it was actually the iconic ship that would have cost the most. It’s important to remember that the Enterprise seen during TNG no longer exists, having been destroyed during the Star Trek: Generations film. Picard, during this time, captains the Enterprise E, coincidentally even more advanced than its predecessor. This version of the ship was technically owned by Paramount, as they owned the rights to the films, while CBS owned and produced DS9. To include the new Enterprise in a way that worked canonically, CBS would have had to pay Paramount a potentially ridiculous amount.
Of course, that doesn't cover the in-universe logic as to why the Enterprise did not partake in the war. As mentioned before, the Enterprise and its crew were the pinnacle of Federation and Starfleet technological advancement. This goes not just for the ship, but also for the crew, being the shining beacons of their fields: they were the brightest and the best (excluding Miles O’Brien, of course). While this produced a killer combination, the consequences of them fighting in a bloodthirsty war was too great for the Federation to risk it. They were much better utilized patrolling Federation space, carrying out diplomatic missions to keep morale up and recruit others to help them. While this might seem like a waste of their potential, the consequences of them being destroyed on the front line would have been disastrous. Imagine the morale of the Federation after learning that the very best they had to offer both in technology and minds, had been unceremoniously destroyed.
It is also suggested in the (technically non-canon) novels that the sovereign class ships such as the Enterprise were powerful enough to hold their own against the Dominion, at least for a while, without the need for full fleet support. This would explain why, if the Enterprise had come up against Dominion forces, it was not mentioned or shown during the DS9 series. They would have been able to dispatch them by themselves without involving anyone else. More like lone wolves, they were able to act autonomously without the need of personal directing orders from those such as the morally ambiguous Sisko, and could be relatively left out of the DS9 narratives.
What sets DS9 apart from the other Star Trek iterations it that it focuses mainly on the sociopolitical atmosphere of a people on the brink of destruction — from the Dominion, but also before this, from various other outside pressures, with little support from the saints in the paradise of Starfleet. Because of this, it felt very different from the stories told in TNG, which is why characters like Q really didn’t work when paired up against Sisko rather than Picard or Janeway. If the Enterprise had played a bigger part during the war and the DS9 series, not only would it have been a bad move tactically and cost effectively, but it wouldn't have made sense theme wise.