The main reasons for the creation of the League of Nations were to stop war, to improve the world, to disarm, and to enforce the Treaty of Versailles. And despite their failure later, the League did manage to stop several wars.
In 1923, an Italian general was killed by Greek partisans while doing work for the League of Nations. Mussolini was angry and blamed the whole Greek nation. He invaded and occupied Corfu, so the Greeks asked the League to help. After the League had a big meeting, they condemned Mussolini and ordered him to leave Corfu and pay reparations to Greece. Their orders had no effect. So the League came up with the solution: Let Greece apologize to Italy and pay them. The Italians left after that. So the League succeeded in stopping a war that would've become very bloody otherwise. This certainly adds to their reputation.
But it was also very embarrassing that Italy, which has signed the Covenant and sworn to peace, would attack another country. If looked at that way, it takes away reputation. So we can deduce that the events in Corfu neither added nor took away from the League's reputation.
2 years after the Corfu accident, some Greek soldiers were killed during a skirmish on the Greek-Bulgarian border. Greece was very angry, so they invaded Bulgaria. Bulgaria asked the League for help. The League summoned the members of the security council and had a meeting. They condemned Greece and told them to leave Bulgaria. At the same time, the Bulgarian ruler ordered soldiers not to fight back, so they won't elevate the severity of the situation. After a while, the Greeks did as the League said and left Bulgaria. And this was another war successfully avoided, thanks to the League. Their reputation...