considering that goldfish should be kept in tanks that are at 68-75 degrees and bettas do better in tanks that are at 80-82, at what temperature have you been keeping this tank?

Goldfish are considered cold water fish. Bettas are tropical.

The eye infection is a result of a bacterial infection. Goldfish produce a massive amount of waste in relation to their size. Three goldfish shouldn't be kept in a tank no smaller than 40-50 gallons.

Most likely the poor water conditions stressed out the betta and thus became more prone to infection.

I would remove the betta, put him in his own tank, and make sure the heater is set at 82 degrees.

Regular water changes/clean water conditions usually will clear up an eye infection like what you've described if you get to it immediately.