While fdb is absolutely correct that the ancient Romans had no conception of language families, we can come up with a plausible calque—a literal translation of each component of a word or phrase.
English "Semitic" comes from German semitisch, "pertaining to Semites", from French sémite, "descendant of the biblical patriarch Shem". So we can analyze the English word as having three components: "the patriarch Shem", "descendant of ___", "pertaining to ___".
The first component is easy enough to translate into Latin, since we just have to look at the Vulgate, where Shem is transcribed as Sēm (via Greek Σήμ).
For the second component, you could use -ānus, as in Julius > Juliānus and Domitius > Domitiānus. Using this for a group of people associated with a particular origin was not uncommon, as in Romānī "Romans" or Christiānī "Christians".
Finally, you could also use -ānus for the third, but applying that suffix twice to the same word just sounds wrong. Instead, I would use the mostly-synonymous -icus (which is also cognate with -ic, -isch, -ικος, etc).
The end result is Sēmānicus, as in linguae Sēmānicae. I think this calque would make enough sense to an ancient Roman, if they were given the context that some populations were thought to be descended from Shem, and people wanted to talk about the languages of those populations collectively.
However, if your audience is not ancient Romans, I would recommend instead just borrowing the English form and using Sēmīticus. It's much clearer to an English-speaker (or a speaker of any modern language, really, since the technical term is widespread). And the -īt- suffix should also be at least moderately familiar to a well-educated Roman, since it comes from Greek -ιτης. It might not be as immediately recognizable as the Latin -ān-, but an educated patrician would be acquainted with Greek nouns like ὁπλίτης ("shield-person", i.e. hoplite) and πολίτης ("polis-person", i.e. citizen).
Also, in post-Classical Latin, this suffix became very popular thanks to the Vulgate; a later Latin-speaker would easily see the parallel between Sēmīta "Semite" and words like Lēvīta "Levite" (someone from the tribe of Levi). While the idea of Semitic languages (or of certain populations being "Semites") wasn't established until much later, the word would be understandable enough to someone in the fourth century.