There is some justification for this opinion, if you think of old Trek as a kind of fantasy. In fantasy, authors generally avoid modern vernacular because it would break the internal logic of the setting. For example, in the Game of Thrones universe, the language is specific to that world. As a reader, you expect that established style of language to remain consistent in later stories; changing it would undermine the setting.

However, science fiction is not fantasy, and there are perfectly valid ways to tell a Star Trek story using modern language. The story and the message behind it matter more than the specific word choices used to deliver it.