We have no real idea, and very little to make any sort of educated guess with. The two main Greco-Roman sources for these events are Herodotus' Histories and Diodorus Siculus' Library of History. The main Jewish sources for the same time period are The Book of Ezra and Josephus' Antiquities of the Jews. As you said, Herodotus references "Syrians of Palestine" and the wars don't appear at all in either Jewish source. Likewise, neither the Jews nor Palestine appear in Diodorus.

That leaves us with a grand total of two sentences in all of ancient literature to work with. Herodotus 7.89.1-2:

[1] The number of the triremes was twelve hundred and seven, and they were furnished by the following: the Phoenicians with the Syrians of Palestine furnished three hundred; for their equipment, they had on their heads helmets very close to the Greek in style; they wore linen breastplates, and carried shields without rims, and javelins. [2] These Phoenicians formerly dwelt, as they themselves say, by the Red Sea; they crossed from there and now inhabit the seacoast of Syria. This part of Syria as far as Egypt is all called Palestine.

Tactically, this lumps "Palestine" in with the Phoenicians as part of the navy. That's not unreasonable since the non-Jewish inhabitants of the region had roughly the same religion as the northern Phoenicians and even the Judeans would have been linguistically and superficially similar as well. There's a potential reference to the Exodus story in there, probably confirming that Herodotus at least knew the Judeans were a dominant group in the region.

It's hard to know how involved the Judeans would actually be in this endeavor though. Judea really only accounted for the inland regions. Tyrian control extended as far south as Joppa and the formerly-Philistine coastal cities of Ashdod, Ashkelon, and Gaza were all politically and culturally independent. However, by this time the Persians had been pulling naval crews from this region for a few decades and its entirely possible that some of the otherwise landlubber Judeans got pulled into the Navy.

Geographically, this includes the sub-province of Judea as depicted in the Jewish tradition, but based on Josephus it would also have included whatever form Edom, Moab, Ammon, Coele-Syria, and Samaria took under the Persians, expanding our modern definition to both sides of the Israel-Jordan border, southeastern Israel, and eastern Lebanon. How these even more obscure (and more inland) people would have played into the navy is hard to know.

However, none of that really matters because as far as Herodotus knew, the Phoenicians and the Syrians of Palestine filled identical roles. Like everyone else in the ancient world, the poor men would be rowers, and the wealthy would be equipped as described above to act as marines and engage in ship-to-ship combat with the Greeks. This means that any Judeans that are alluded to in this one passage would have only participated in two major engagements during Xerxes' invasion: Artemisium and Salamis, where they would have participated in trying to incapacitate and possibly board and capture Greek ships. Artemisium was something of a draw, with the Greeks making a strategic retreat after their land-defeat at Thermopylae, but Salamis was a bloodbath for the Persian side that left the fleet permanently crippled.

After his catalog of nations in Book 7, Herodotus never mentions the "Syrians of Palestine" again and just refers to the much larger Phoenician contingent, but if we assume they were still lumped together, then any Judean survivors of Salamis would have had the mercy of being sent home before the second crushing "naval" defeat at the Battle of Mycale (which was actually fought on land).

Since we have no other mentions of Judeans or "Syrians of Palestine" participating in the Greco-Persian Wars, it's impossible to know if they were normally included under the "Phoenician" label or not. According to Herodotus they were visually identical and most Greeks probably couldn't have easily identified their languages that precisely in combat so it is possible. If they were lumped in as Phoenicians, then it's entirely possible that at least a few Judeans were involved with naval warfare against Greeks of one sort or another at different times over the course of more than 30 years between the Ionian Revolt and the Battle of the Eurymedon.