It's pretty safe to say that the whole "Batman works alone" myth has been thoroughly debunked. You don't have to be the world's biggest comics fan to know about the Robins or the Batgirls or the various members of the GCPD and extended Bat-Family who regularly move in and out of Gotham, helping the Dark Knight with his crusade. No, Batman does not work alone, and each of his partners, sidekicks and associates brings something unique to the table. But one, perhaps above all others, tends to bring just a bit more.
It's not a competition, and it's certainly not an objective metric, but Alfred Pennyworth may just be the single most indispensable component in Batman's vast support network. And he does it all from the relative safety of Wayne Manor, wearing no mask, carrying no weapons and with no flashy costume or emblem to call his own.
Alfred's story actually begins pretty oddly. He didn't start out as the tall, wiry, ex-special forces, Shakespearean actor we know today. In fact, he didn't start as much of anything but a background character a full sixteen issues into Batman's original solo series in 1943. You definitely wouldn't recognize him in those early days—he was a portly, bumbling detective who looked more like a caricature of Sherlock Holmes than anyone's butler. "Caricature" was really the name of the game, too. This was an Alfred who was specifically designed to make Batman and Robin look all the more skilled as detectives by fumbling his way through virtually every case.
This incarnation didn't last very long, but the updates didn't start in the comics. By late 1943, the original Batman live-action serial was playing in movie theaters across the country featuring an Alfred played by actor William Austin, a tall, thin man with a pencil moustache. Austin's portrayal was such a hit that Alfred's look was given an overhaul in the comics less than a year later with the in-fiction reason for the change being that he "took some time off at a health resort," slimmed down and tried something new with his facial hair. No, really.
Over time, Alfred's backstory was slowly filled in, establishing him as the son of Jarvis Pennyworth, the Wayne family's previous butler. After Jarvis' death, Alfred had applied for a job as Bruce's "valet," only to eventually be upgraded to butler. His background began solidifying as well, including his acting skills and his military service. It didn't happen overnight by any means, but eventually—roughly around the mid-1950s—Alfred was pretty recognizable to modern eyes.
Once his presence in Gotham really began to set, Alfred rapidly became quite a jack-of-all-trades. His acting prowess became a quick and easy way to explain different disguises Bruce needed, and his military and medical training was a great way to handle various injuries sustained without having to show Batman going to a hospital or a doctor while in costume. Alfred was a key component in maintaining Bruce and Dick's secret identities as well, often running interference between the private and public aspects of their lives when things got a bit dicey.
Alfred proved himself so indispensable to the foundation of Gotham's crimefighting community that he was even allowed to return from death. In the 1960s, during a major downswing in the comics market, Alfred was (rather unceremoniously) killed off in DETECTIVE COMICS #328 to try and bring a new spark of life to the Bat-Books. He was replaced with a new "caretaker" character, Aunt Harriet, who only existed for about two years before Alfred was surreptitiously returned to both life and his post. He'd go on to make it through more major cosmic upheavals than you can count, including CRISIS ON INFINITE EARTHS, which effectively rewrote the history of the DC Universe from the ground up.
Of course, that doesn't mean that Alfred's totally invincible. He may not be on the front lines fighting with Bruce and the Bat-Family, but he's always in his fair share of danger. Just take a look at the end of the New 52 and BATMAN: ENDGAME, which sees Alfred suffering some pretty extreme losses as he fights to keep the city from crumbling at the hands of the Joker.
Don't worry, he does get better.
Historical context and actual utility aside, it's important to note that Alfred is also just an important part of Bruce's life. Alfred's place in the big picture of the Dark Knight is as crucial, if not more so, than that night in Crime Alley and the string of pearls. It may not seem like it all the time because it's so easy to take what he does for granted, but a Bruce without an Alfred would be disastrous on the most fundamental level—I mean, have you seen Bruce when he starts moping super hard? Have you experienced the level of melodrama he's prone to when things aren't going his way?
Without Alfred there to kick him into gear every few weeks, we wouldn't have a very functional Batman at all. In fact, we probably wouldn't have a Batman, period. For all Bruce is motivated by his quest for vengeance and the trauma of his past, his ability to maintain a perspective on himself and his crusade is what prevents him from completely going off the deep end time and time again. He's not a hero because he's angry, he's a hero because he's able to turn that anger into something more important and much larger than himself, and that wouldn't happen without Alfred there as his infallible moral compass.
So, sure. Batman Day might be a literal celebration of all things Batman. (You know, like the name implies.) But take some time out of your busy schedule to appreciate the things that make both Bruce and Batman who they are. Alfred may not be a literal caped crusader, but he is absolutely a superhero.