It depends on continuity.

• Originally, he served as a comic relief. In most early tales, he made bungling attempts to be a detective on a par with the young masters.(wiki)

• Pre-crisis, he was a trusted support staff. NOT on "could not manage being Batman without" level, however.

• Post-crisis (after 1983), his role was much bigger. After the death of Bruce's parents, Alfred was appointed his legal guardian and became a father figure, raising Bruce to become the man that became Batman.

Aside from serving "merely" as a butler, Alfred also serves as :

• Helper when science and engineering is concerned. He maintains equipment such as BatComputer. Especially in Pre-Nolan movies, he pretty much seems to assume Lucius Fox's role.

• An expert on first aid and medicine. Without him, Batman would risk detection much more due to having to go to the hospital.

• At times as field helper. He is frequently portrayed as a capable combatant. In many versions, he's ex-special-forces.

• He is also serving as archetypal "wise old man", giving Bruce advice, helping him keep his moral compass, serving as a psychologist, sounding board and the only person Bruce can trust.

It depends on continuity.

• Originally, he served as a comic relief. In most early tales, he made bungling attempts to be a detective on a par with the young masters.(wiki)

• Pre-crisis, Alfred Thaddeus Crane Pennyworth was a trusted support staff. NOT on "could not manage being Batman without" level, however.

• Post-crisis (after 1983), his role was much bigger. After the death of Bruce's parents, Alfred was appointed his legal guardian and became a father figure, raising Bruce to become the man that became Batman.

Aside from serving "merely" as a butler, Alfred also serves as :

• Helper when science and engineering is concerned. He maintains equipment such as BatComputer. Especially in Pre-Nolan movies, he pretty much seems to assume Lucius Fox's role.

• An expert on first aid and medicine. Without him, Batman would risk detection much more due to having to go to the hospital.

• At times as field helper. He is frequently portrayed as a capable combatant. In many versions, he's ex-special-forces.

• He is also serving as archetypal "wise old man", giving Bruce advice, helping him keep his moral compass, serving as a psychologist, sounding board and the only person Bruce can trust.

Your last question "Could Batman manage being Batman without Alfred?" doesn't really have an objective answer. May be he could, may be not. It most certainly would have been more difficult for him.