The Mirror reports that thousands of viewers complained on the BBC website, while forum and chatroom members said that Dixon does not have enough experience for the job.
The singer reportedly asked for help from her mother and grandmother, who were sitting behind her during the first show on Friday.
"She was also seen asking fellow judge Bruno [Tonioli] for his advice," an audience member said. "They scored three of the eight dances the same and four of their scores were only one mark apart.
"When the dancers were being interviewed backstage the panel were off-camera. Alesha turned around in her chair between almost every dance and asked her family their view.
"We could see her mouthing, 'What do you think?' then asking for a thumbs-up or thumbs-down. I also heard her asking Bruno what he thought several times."
A BBC spokesperson said: "We're pleased at how fantastically well Alesha was received for her debut on Strictly Come Dancing last night and delighted that so many people tuned in and enjoyed the show."
The decision to replace judge Arlene Phillips, 66, with 30-year-old Dixon caused controversy earlier this year as fans complained that the BBC was ageist.