The Delian League was an alliance of Greek city states set up in 477 BC in the aftermath of the defeat of the invasion of Greece by Xerxes in 480 BC - 479 BC.Its purpose was to provide for a common Greek defence against any further invasions.However Athens - leader of the League and by far its most powerful member, used the forces and resources of the league for its own ends,and essentially bullied all the other members into a position of subservience.This meant that the Delian League effectively became an Athenian empire rather than an alliance.
Why did its structure and membership weaken the League?
Assuming you mean the League of Nations:
Major powers weren't members from the beginning.
USA opted out at the beginning and was never a member,,Russia,which had become the world's first Communist state,wasn't initially invited,and didn't join until 1934 (as the USSR),and Germany wasn't allowed to join until 1926.Japan and Germany both left in 1933,Japan in protest at league criticism of its invasion of Manchuria,and Germany as soon as the Nazis came to power (Hitler saw the League as an instrument of the Treaty of Versailles,a treaty which he wanted to overturn).The USSR was expelled late in 1939 for aggression against Finland,and Italy had left in 1937
All this left Britain and France as the only 2 major world powers that remained members throughout of what was supposed to be a global organization.This gave them a lot of influence which they often used in their own interests;when the League wouldn't bend to their will,they ignored it and made deals that bypassed the organization,but were unwilling to use force to back up the League as an international organization.
The League was thus inherently weak from the very beginning and throughout its existence.