The dialogue in B5 seems like a dated screenwriting style because it is. Most of what's dated is the humor--the gags and character wisecracks fall into a set of 90's humorous screenwriting formulas, just as the wisecracks and gags in say, "Star Trek: Discovery" fall within today's set of humorous screenwriting formulas. (There are some humorous bits in B5 that are still funny, like the line where Londo calls Vir a "moon-faced assassin of joy", but most of these work because they sound like something the character would actually say and not just a cheap screenwriter joke formula.)

But some of the writing still holds up.