The question is asked given that there is a real risk for injuries (e.g. broken ribs, concussions). Also, it is in the context of recreation/health/fitness rather than a profession.
In a somewhat related question (Is sky-diving forbidden because of the risk to one's life?), I showed that normal risky activities (as defined by whether other members of society accept them as reasonable, or whether the risk of danger rarely occurs) are permitted.
As such it appears boxing would be technically permitted, and R Isser Unterman (Chief Rabbi of Israel) is quoted (here) as permitting boxing and other forms of combat sports, arguing that they do not fall under the category of hitting with malice.
However, not everything permitted should be done, and many rabbis came out against boxing since it is a violent sport with a goal to hurt others on purpose. This article brings many relevant quotes (it relates to watching boxing but applies in many ways to being in the ring)
[Boxing] certainly has a negative effect on one’s middos and does not lead one to be the fine, sensitive, kindly human being the Torah expects one to be -- R Ben Zion Shafier
Boxing elicits cruelty within a person. Additionally, hitting another Jew is prohibited, which means that watching boxing involves watching something that’s prohibited at least to Jews -- R Zev Leff
I wouldn’t recommend watching or enjoying such a “sport.” It leaves a stain on one’s soul -- R Marc D. Angel