Yes, Steve Harvey's career will likely suffer a bit after his mistake as host. While the mistake is an understandable one about which most previous hosts have probably had nightmares, a mistake is a mistake. It will be hard for others to select Harvey as host for their programs. In the long-run, however, Harvey has a broad career that extends into other areas so he probably will not suffer greatly.
It was an honest mistake. Everybody makes one and just because he's famous doesn't mean he should be perfect. It was an accident and he apologized. I don't think his career will suffer. He might not be hired for that job again, but it won't affect his show or anything he does outside of the show.
Steve Harvey is a well respected talk show and game show host. I do not think his career will suffer from announcing the wrong winner at the Miss Universe Competition. Although it was a huge and horribly embarrassing mistake, he recovered with great dignity to quickly and profusely apologize. This mistake will be media fodder for a while, and I'm sure will be made fun of by every late-night talk show host,in the long term, I do not think it will hurt his career.
Although his misreading of the winner was a shocking mistake to both communities, it was just that: a mistake. He immediately sincerely apologized on several media platforms and took full blame for the mistake. He reminded everyone that the focus should be on the contestants and sincerely apologized multiple times.
Even though Steve Harvey made mistake and named wrong winner, I don't really think that his career will suffer. Sometimes such mistakes happened, but it doesn't mean that he is unprofessional or incompetent. He said that he apologizes for this and didn't want to abuse any of participants, I think it's enough to close the episode.