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The term semiticus is attested in Rudimenta linguae Hebraicae (C. H. Vosen, 1883) but I am more interested in Classical Latin.
In English, the expression "Syro-Arabian languages" is sometimes used but I don't know if such a compound is of good latinity.
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.
The ancient Romans had no concept of language families, nor of Semites (the supposed descendants of Noah’s son Shem). There is consequently no word for “Semitic languages” in Classical Latin.
As pointed out, the Romans did not have a concept of language families as we understand them today. They did however have a concept of some peoples speaking languages that were similar to or "the same" (presumably close enough that the author, who may not have understood either language well, could not distinguish them) as other languages they were more familiar with.
The Semitic-speaking people the Romans would have had the most familiarity with would have been the Carthaginians (note that Punic remained in use in parts of Africa, especially Tripolitania, right up until the Muslim conquest, and there is evidence of people being taught it as a classical language in Rome even after the Punic wars), probably followed by the Syrians (who would have spoken various Aramaic varieties).
With that in mind, I suspect that they would have referred to Semitic languages as "sermōnēs similēs Pūnicae" "languages like Punic" or "sermōnēs similēs Syricae" "languages like Syrian".
It's unclear which languages they'd have recognised as fitting which labels though.
Punic was much less conservative than Aramaic in this era, having lost the gutturals (retaining only a single /h/ mostly from earlier /χ/) and possibly merged most of the sibilants (communis opinio is that this definitely occurred, but I'm skeptical as the argument frequently relies on evidence that couldn't actually distinguish either way). It did still maintain a three-way distinction between the voiceless, emphatic, and voiced stops although it's possible the emphatics had lost their glottalisation or pharyngealisation. As the Greeks and (educated) Romans seem to have perceived the emphatics' lack of aspiration (which was present on the voiceless stops) as their distinctive characteristic though (and this is certainly preserved) this likely isn't a particular concern.
We know that Punic & Hebrew were recognised as closely related in late antiquity as St Augustine leans heavily on his native Punic in interpreting scripture. Prior to the complete displacement of Punic script with Latin for writing it the orthography was also extremely close to Hebrew so that even if a Carthaginian and Judaean would struggle to understand each other's speech, they wouldn't have too much difficulty in writing (noting that knowledge of Paleo-Hebrew, which would have had clear parallels with Punic script, unlike the square Aramaic script, does seem to have continued into the Roman era as Hasmonean coinage frequently contains it).
Similarities between Punic & Aramaic would probably have been much less obvious, although those between Aramaic and Hebrew would also likely have stood out. As such I suspect that Hebrew could reasonably have been described in either of these ways, but that Hebrew (and Phoenician-proper) would likely be the only languages identified as similar to Punic.
I suspect that Akkadian (which was retained as liturgical language in Mesopotamia into the Common Era) would not have been recognised as related. The Arabic of the Nabataeans, Palmyra, and Hatra probably would be, albeit on fallacious grounds (these peoples all used Aramaic for their inscriptions, which could easily be recognised as akin to that of the Syrians, and the Arabic actually spoken by their peoples may have been taken as a debased form of this Aramaic, rather than an independent, but related language).
I suspect other Semitic languages would lack enough obvious similarity for the Romans to have realised they were related.
