Another indicator of that seriousness is that the team has already banned a fan for the use of racist language during a game. Here's what the team said in a statement Wednesday:
"During Tuesday night's Red Sox-Orioles game, it was reported to Red Sox security that a racial slur was used in a comment from one fan to another fan. The offending individual was promptly ejected from the ballpark and has since been notified they are no longer welcome at Fenway Park.
"The Red Sox organization will not tolerate the use of racial slurs at Fenway Park, and we have apologized to those affected. There is no place for racial epithets at Fenway Park, in baseball, or in our society. The Red Sox have turned the matter over to the Boston Police Department, who will further investigate with their civil rights unit and determine whether it merits further action."
The phrasing "no longer welcome at Fenway Park" certainly sounds tantamount to a lifetime ban, so it appears that the policy the team was considering is already in effect, at least in this particular incident.
To preempt the usual objections to this sort of thing, recall that the Red Sox are a private entity, and private entities are not subject to the First Amendment.