Why is the secret Rebel base so important?

Question

Why is the Empire is so concerned with finding the Rebel base? If the Empire controls thousands or millions of planets, would one base with a few thousand Rebels matter even a little bit?

If the Rebellion is a threat at all, I assume it would be because there are rebel factions on thousands of planets, and/or that hundreds or thousands of systems have “slipped through” the Empire’s fingers and are actively resisting it, perhaps with their own local armed forces. If so, one Rebel base should not matter, and destroying it would probably do nothing to slow down the Rebellion. Similarly, it’s hard to believe that the Rebel leadership would be so concentrated that the Rebel Alliance couldn’t survive the loss of its headquarters.

Is this simply a case where the Empire’s leaders are determined to fight a conventional war and are fooling themselves into thinking that destroying the secret Rebel base will somehow defeat the Rebellion? Or are the secret Rebel bases in Star Wars: A New Hope and The Empire Strikes Back actually important for some reason?

Answer 1

Not sure if you are asking about episode IV or VII? I'll answer all 3 just for giggles.

Episode VII

Alan Dean Foster's TFA novelization explains that, straight from the horse's... errr Supreme Leader's mouth: it was all about a cute astromech droid and the info it carried:

Collected and composed as he was, Hux was not immune to surprise. “The system? Supreme Leader, according to the most recent galographics, at least two and possibly three habitable worlds circle Ileenium. Following the destruction of the Hosnian worlds, would it not be worthwhile simply to destroy their base and claim the remainder for the Order? We will have the location of the base within a matter of hours and—”

Snoke cut him off. “We cannot wait. Not even for hours. Hours that may permit as little as one ship to depart with the information that will allow them to find Skywalker. That would be one ship too many. The more time we give them, the more likely the chance,

Episode IV

A vast bulk of Rebel forces were on Yavin. That was because it was a fortres and main military base for them

• Tarkin (and Vader) explicitly said so:

  After a suitable interval had passed, he motioned to the machine. “Now, Senator Organa, Princess Organa, we will discuss the location of the principal rebel base.”

  ....

  “Lord Vader will provide us with the location of the rebel fortress by the time this station is certified operational,” Tarkin declared. “That known, we will proceed to it and destroy it utterly, crushing this pathetic rebellion in one swift stroke.”

  and

  Tarkin turned to him, nodding. “The Senate is being informed of our action at this very moment. Soon we will be able to announce the extermination of the Alliance itself, as soon as we have dealt with their main military base. Now that their main source of munitions, Alderaan, has been eliminated, the rest of those systems with secessionist inclinations will fall in line quickly enough, you’ll see.”

  ... and yes, if you're thinking that this was in some way a subtle nod to Kurosawa, you're wrong. it was an UNsubtle, category 11, shoutout to Kurosawa:

  “Fagh! You just said it yourself, Vader: we’ll get nothing more out of her. I’ll find that hidden fortress if I have to destroy every star system in this sector. I’ll—”

• The reason was because the ancient culture who built giant stone-blocked temples that were excellent protection:

  While the science of this moon’s inhabitants had led them to a dead end as far as offworld travel was concerned, they had produced several discoveries which in certain ways surpassed similar Imperial accomplishments—one of which involved a still unexplained method of cutting and transporting gargantuan blocks of stone from the crust of the moon....

  Theoretically, no weapon could penetrate the exceptionally dense stone of the ancient temple, but Luke had seen the shattered remains of Alderaan and knew that for those in the incredible battle station the entire moon would present simply another abstract problem in mass-energy conversion.

This "bulk" included most of their high level political backers

... General Jan Dodonna adjusted the tiny mike on his chest and indicated the small group seated close to him.
“You all know these people,” he intoned with quiet power. “They are the Senators and Generals whose worlds have given us support, whether open or covert. They have come to be with us in what may well prove to be the decisive moment.” He let his gaze touch many in the crowd, and none who were so favored remained unmoved. (ANH novelization - By Alan Dean Foster)

Episode V

They couldn't care less about the Rebel Base. Vader was searching for Luke Skywalker. The Rebels were merely a useful side benefit to that.

Captain Piett respectfully gazed at his master, who loomed above him like a black-robed, omnipotent god. “Yes, sir,” Piett said slowly, choosing his words with caution. “We have visuals. The system is supposed to be devoid of human forms …”

But Vader was no longer listening to the captain. His masked face turned toward an image beamed on one of the viewscreens—an image of a small squadron of Rebel snowspeeders streaking above the white fields.

“That’s it,” Darth Vader boomed without further deliberation.

“My lord,” Admiral Ozzel protested, “there are so many uncharted settlements. It could be smugglers—”

“That is the one!” the former Jedi Knight insisted, clenching a black-gloved fist. “And Skywalker is with them. Bring in the patrol ships, Admiral, and set your course for the Hoth system.” Vader looked toward an officer wearing a green uniform with matching cap. “General Veers,” the Dark Lord addressed him, “prepare your men.”

(Donald Glut's ESB novelization)

Answer 2

This one sentence—written by George Lucas in 1973—sums up the basic conflict of Star Wars:

“A large technological empire going after a small group of freedom fighters.”

To answer a question like this you really need to look at George Lucas’s out-of-universe inspiration for the whole Star Wars saga in the 1970s: The U.S. war in Vietnam. As explained in Chris Taylor’s How Star Wars Conquered the Universe; bold emphasis is mine:

Lucas was determined now to make movies about war in three modes: past, present and future; absence, reality an allegory. American Graffiti would take people back to a time before Vietnam ripped America apart. Apocalypse Now, which Lucas hoped to direct before or after The Star Wars, would show it in the present tense. If THX was the movie he expected to get him banned from Hollywood for life, Apocalypse Now was the movie he felt wild lead to the government running him out of the country.

The third mode that Lcase intended to use to depict Vietnam—the allegorical futuristic lens—was only just taking shape, but already it was being influenced by Lucas’s thinking about the present tense. Lucas was fascinated by the notion of how a tiny nation could overcome the largest military power on Earth, and this was baked into The Star Wars right from its earliest notes in 1973: “A large technological empire going after a small group of freedom fighters.”

So knowing that, a lot of the obsession one see’s with villains in the Star Wars universe is based on the arrogance of technical superiority and the obsession of control ultimately being the downfall of the villain more so than the good guys fighting back stronger.

To look at the Vietnam example, the U.S. was trying to stop the philosophy of communism spreading. With the mindset that if China’s Communist philosophy spread across Southeast Asia somehow Democracy would collapse around the world. In my opinion, it’s an utterly paranoid nonsense idea. That somehow if you attack a small group of people who are philosophically opposed to you that you somehow will stop some greater “bad” from happening.

In the case of Star Wars the bad guys are just seen as paranoid control freaks. You’re right, the Rebels are opposed to the Empire so who should care in the end… But that assumes the bad guys in Star Wars are acting rationally when they are not and have never. The irrationality of the Empire—and now the First Order—in their attempts to control what is not controllable is what destroys them in the end.

Also, if you are confused as to why the Empire was obsessed with hunting down the Rebels, think about artists and poets in our real world who simply create images and write words to only be censored, banned or even killed by fascist regimes.

What exactly is so terrifying to a military power about one artist speaking their voice? Easy: The spread of ideas that spread the concept of individuality and personal self-expression terrify people who demand control. Why did the Nazis plunder artworks in World War II and destroy—and sometimes the artists—them as well? Why did the Khmer Rouge slaughter musicians and artists in Cambodia? Same mentality.

Control freaks hate freedom and individual thought.

That said, this basic concept that spurred George Lucas’s creation of Star Wars has been watered down to death over the years. But sometimes the overarching message that folks as plain and simple as a farm boy (Luke Skywalker), a smuggler (Han Solo) and his Wookiee, a gambler (Lando Calrissian) and the Ewoks can somehow aid in the defeat of a massive technologically superior foe like the Empire still comes through.