Yes, they had special horse transports, which were converted from wide-bodied, sail-powered merchant ships. According to Thucydides, they were invented in Athens in 430 BC:

On board the ships he took four thousand Athenian hoplites, and three hundred cavalry in horse transports, then made for the first time out of old ships; fifty Chian and Lesbian vessels also joining in the expedition.

-- Thuc. 2.56.2

From a later passage (Thuc. 6.43) we learn that one transport could carry 30 horses (though we don't know if this was a full load, or if horse transports were made to a standard size).

If Thucydides is right, there were no specialised ships for this purpose in the Greek world before the Peloponnesian War. Indeed, we hear of no earlier Greek overseas expedition in which cavalry was brought along (unless we count the mythical Trojan War). Horses that were being traded or shipped with some rich man's retinue would likely have been moved in ones and twos in the hold of merchant ships. Cavalry could not be shipped in significant numbers until this Athenian innovation.

However, by this time, the concept was already well known elsewhere. In preparation for the campaign that would end at Marathon in 490 BC, Darius ordered the Ionians of the coast of Asia Minor to build triremes and horse transports (Herodotos 6.48.2). Many such ships also came along with Xerxes when he invaded Greece ten years later; Herodotos seems to have considered them one of the noteworthy marvels of the expeditionary force (7.21.2, 7.97).

If it was indeed the Ionian Greeks who built them for the Persians, it is odd that they didn't adopt these ships themselves until 50 years later. Possibly there had been few calls for a ship that could carry cavalry overseas; their services were hardly required, for instance, in the Delian League's campaigns against islands and coastal strongholds. Either way, the Athenian "invention" of these ships should actually be recognised as a late adoption of a tried and tested idea. Thucydides probably made it out to be more revolutionary to reinforce his general point that the Athenians were restless and ambitious innovators. (Alternatively, we may be reading him wrong, and he may only have meant to say that this was the first time old ships were converted into horse transports, rather than being purpose-built as such.)