Okay, so this is actually a pretty interesting question because the Gothic Kingdom of Spain was in a prime position for naval activity in the western Mediterranean, yet we hear almost nothing of it.

Seaborne raiding in late antiquity was something that happened, although probably not as often as most think, and not in the way that most think. Modern images of the Saxons typically portray them in "Viking-style" boats being intercepted by Roman riverine patrol vessels or liburnae, which is far from the reality. In fact we have almost no evidence for the presence of widespread naval activity in the North sea, all of it points instead towards the Romans focusing on a transport system because it was nearly impossible to patrol the coasts with their technology, let alone intercept raiders. Hence the need for a series of fortifications and watchtowers called the "Saxon Shore," rather than an extensive naval defense system. Almost all naval activity in this period was really transport for land-based raiding, and there were few "naval battles" outside of Roman internal conflicts. But let's take a look at what we do know.

The various Gothic peoples living north of the Black sea launched seaborne raiding campaigns several times in the 3rd Century AD, most notably in 255-258 and 275, where they penetrated the Straits of Marmara and raided all across the eastern Balkans and much of the Anatolian Coast. The problem is, it's hard to say how or why in the 3rd Century AD that seaborne raiding across the Black sea suddenly became prominent. There's not enough information from our primary sources reporting on this period, so we can only guess. Some sort of socioeconomic pressure had drove them into the Crimea where they were harassing and captured many cities of the Bosphoran Kingdom, which likely facilitated this raiding. We don't know if they had their own clinker-built ships like the peoples of the Danish bogs at Nydam, Ejsbol, and other sites along the North Sea and Baltic coasts. But the Bosphoran Kingdom and the many cities in the North Pontic region were on the end of the northern routes of the "silk road", and there were many merchant ships and shipwrights that could have potentially been made available by their capture. This probably facilitated the temporary increase in seaborne raiding seen in the mid-late 3rd century. The Romans did counter this with increased patrols, but it was ultimately a land battle in 276 that ended Gothic raiding in the region.

The Visigoths, however, would not come into being for a century and a half after these events. First really achieving ethnogenesis some time around the death of Alaric, they would end up being... "settled" (that's a really complex argument, long story short, about how Barbarian Foederati were "settled" in the Roman Empire), in the Roman provinces of modern Aquitaine in 418 AD. They basically were more like administrators, bureaucrats, and semi-professional soldiers than they were migrating refugees, and were more like a semi-independent client bureaucracy than a kingdom. But I digress.

Study of naval activity in the Mediterranean is mostly focused on the Vandal invasion of Africa in 429-439 and their subsequent raiding of South Italy, Corsica, Sardinia, and Sicily. It's also one of the few instances where an actual naval battle is recorded, in 457 or 458 off the coast of Corsica (dating is a bit shoddy for things from this period), where Majorian's Magister Militum Ricimer defeated a small raiding fleet. But like earlier periods, the Vandals were mostly just utilizing Grain ships as transport ships (or in the case of the Battle of Cape Bon, fireships) and most of the piracy and raiding was land-based.

So where do the Goths come in? Like the Vandals, the Goths had some access to Roman merchant ships and shipwrights when they eventually moved into Spain. Prior to about 456 or 457 AD, the Visigoths had absolutely no access to the Mediterranean. Their attempts at gaining leverage and concessions from the Romans by besieging the Mediterranean port cities of Narbona and Arelate (Narbonne and Arles) from the 420's-450's had been consistently thwarted by Flavius Aetius, who kept the Visigoths confined to the terms of their treaty. However, with the ascension of Avitus the Visigoths seem to have finally been put on the same standing as the Vandals, and recognized as an independent power with the status of socius et amicus, like Gaiseric had attained in the Treaty of 442. The Goths would from that point forward slowly start expanding into Northern Spain, although they were briefly reined in by the emperor Majorian in 458-460. This gave them access to the Mediterranean ports, Mediterranean trade, and Roman shipwrights, much like the Vandals had used to cross over into North Africa.

But we see very little evidence of Gothic naval activity in the 5th century. There is one exception: Hydatius' chronicle in entries 171 and 194 records that a force of Heruli attacked the Suebes in Gallaecia. This is interesting because there has long been an argument that there were two groups of Heruli, but recent scholars have reassessed that and the notion of a group of Heruli living along the Rhine has been rather thoroughly debunked. Instead, it seems far more likely that these Heruli were part of the Visigothic kingdom, having migrated from the Pontic region either alongside them or at a later point. They may also have been in the employ of the Romans, but both hypothesis are about equally plausible. This is one of the few recorded instances of potential Visigothic naval activity.

The 6th Century also shows signs of similar activity by the Visigothic Kingdom. Before 533 the Goths appear to have taken the fortress of Septum (Ceuta) from the collapsing Vandal Kingdom, probably as a Bulwark against the Roman reconquest. The Romans took Ceuta in 533, but in 540 the Goths again sent an army and took it back, only to again be retaken by the Romans. The Visigothic Kingdom also sent a force of 500 men all the way from Spain to Carthage in 697-698, to help the Romans retake and defend the city against the Umayyad Caliphate. This is indicative that the Goths were capable of sending armies around the Mediterranean, but we simply lack information on whether they had any capacity to fight a naval engagement, and the likelihood is that they did not. They do not seem to have had proper late Roman warships or the capacity to build vessels similar to the military Dromon-type ships fielded in the 5th-7th centuries, unlike the Romans and the later Arabs.

So I hope this helps answer your question. In summary, the Goths had some capacity to transport troops, mostly through seized merchant vessels, but they seem to have lacked a formal navy.

Sources:

• Edwards et. al. The Cambridge Ancient History, Volume 12: The Crisis of Empire, AD 193-337. (Cambridge: Cambridge University Press, 2005).

• Roland Steinacher, "The Herules: Fragments of a History," in Neglected Barbarians (Turnhout: Brepols Publishing, 2010).

• Roger Collins, Visigothic Spain: 409-711. (Oxford: Blackwell Publishing, 2004).

• Harold Livermore, Twilight of the Goths: The Rise and Fall of the Kingdom of Toledo, 565-711. (Bristol: Intellect Books, 2006).

• Hydatius, Continuatio Chronicorum.

• Prokopios, The Wars of Justinian, Trans. Anthony Kaldellis.