She looked ridiculous during the performance, and didn't handle it very well. However, she still has one of the best voices, and has sold millions of records, so this won't kill her career. The fact that companies like DIck Clark entertainment are still trotting her out on NYE and other holidays to 'sing' well, really perform, her songs is proof of her longevity.
New Year's and Super Bowl half time shows can be career killers or savers for artists today. Mariah's fail is sure to hurt her reputation and will negatively impact her career. It may not be the end of her career but it will take some time for her to get past this.
The people who are Mariah Carey's fans do not care if she lip syncs or not. They probably already suspected it. They are the ones willing to pay money to see her, rather than just watch her for free on television. They are not bothered by her having one bad day. They probably want to tell her not to let it bother her.
No, Mariah Carey's New Years lip-sync fail will not hurt her reputation. Her reputation as a singer, as a performer, and as a diva was solidified long ago with her fans and in general. Her fail and more importantly her reaction afterward are simply further evidence of her attitude, which no one is oblivious to in the first place.