Basel, under interim coach Alexander Frei, were thumped 3-0 at home by PAOK on Wednesday in the second leg of their second round qualifier and lost 5-1 on aggregate.
Koller, who stepped aside as Austria coach last year after six years in charge during which he led them to Euro 2016, has previously won the Swiss title with St Gallen and Grasshoppers. He has also coached FC Cologne and VfL Bochum in the Bundesliga.
"We wanted an experienced coach with a strong personality," said the club's sporting director Marco Streller.
Koller replaced Raphael Wicky who was sacked one week ago following a 2-1 defeat in the first leg against PAOK.
Wicky was two games into his second season at Basel.
His first season saw them fail to win the Swiss league for the first time in nine years, although they reached the last 16 of the Champions League.
(Writing by Brian Homewood; Editing by Toby Davis)
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