The energy drink company, which owns Dortmund's Bundesliga rival RB Leipzig along with teams in Salzburg and New York, made the surprise announcement on Wednesday, with the former Liverpool coach starting his role in January.
Klopp became a local hero in Dortmund, delivering two Bundesliga titles and a German Cup during his seven years at the club while also taking them to the 2013 Champions League final.
The Dortmund fanzine Schwatzgelb, however, said that Klopp had destroyed his reputation and "torn it down with a bang".
It also took issue with Klopp's self-label of 'the normal one', a tongue-in-cheek reference to Jose Mourinho's declaration he was 'a special one' on arriving at Chelsea.
"The normal one, unfortunately (is) normal on all levels. Normal like everyone else in this dirty business," the magazine wrote.
"Jurgen Klopp's decision to support Red Bull has ruined his life's work, at least for a large proportion of fans in Dortmund and in Germany," it said while adding he had given a "well-mannered middle finger" to Dortmund fans.
He had retained his reputation in Dortmund in spite of leaving for Liverpool in 2015 where he went on to win several titles, including the Champions League and Premier League, before stepping down in the summer, citing a lack of energy.
He was given a rousing reception at a testimonial match in September in front of 81,365 fans at Dortmund's Westfalenstadion.
His willingness to speak out on social issues - including commercialism in football - has also formed part of his legacy, as has his 'everyman' appeal.
In England, he spoke out against match time slots being determined by TV networks as well as the broader issues of populism and Brexit.
He was openly critical of the European Super League in 2021, despite support from Liverpool's ownership for the breakaway competition.
His decision to take up the job at Red Bull has annoyed Dortmund supporters who, like many fans in Germany, have a long-standing dislike of RB Leipzig.
Some fans believe Leipzig, founded in 2009, are not consistent with Germany's 50+1 rule, which requires member control of clubs, while others have hit out at the multi-club ownership model used by Red Bull.
'No surprise'
Surprise at Klopp's decision was not uniform, however, even at Dortmund where managing director Hans-Joachim Watzke wished him "good luck in his new job".
"It was no surprise to me because we talked about it personally," he told AFP's German sports subsidiary SID on Wednesday.
Klopp had previously defended the Red Bull model while calling himself a football "traditionalist".
In an interview with the DFL, who organise the Bundesliga, Klopp said he "understood how RB Leipzig was criticised by football traditionalists - of which I am one.
"But I find that Leipzig hasn't taken anything away from traditional clubs, they've just gone their own way.
"They don't earn more money than Dortmund, or Bayern. They develop young players - the whole idea is a football idea and not a money idea."
Former Manchester United coach Ralf Rangnick, who held the same position Klopp will take over in January, said: "It will certainly be exciting to see what Jurgen can contribute."