The first incident happened on the Rock & Roller Coaster when two cars of the rollercoaster collided at the start of the ride, said news agency ATS.
The braking system failed for an unknown reason and the second car crashed into the first, the Basel-City public prosecutor said.
One woman was taken to hospital and four other adults were treated at the scene. Two children in the affected cars were unhurt.
An investigation is underway to establish why the brakes failed, with parts of the mechanism taken away for examination.
The ride was temporarily shut down but reopened on Sunday night after safety tests, reported ATS.
A second accident happened ten minutes after the first on the Round up Rounder carousel, when a man decided to leave the ride while it was running. He was thrown to the ground and seriously injured.
Accidents at Basel's annual autumn fair are rare.
The event, the oldest fair in Switzerland, kicked off on Saturday and features around 50 rides and 500 stalls.
More than a million visitors are expected to attend the event, which ends on November 12th.