I'm sorry I don't have time for a more detailed answer, but I think that this question partly stems from the idea that “Visigoths” (in AD 398) actually were an entire people on the move, something like a group of refugees. This is a much-debated issue, but I am of the opinion that we ought to conceive them as something more akin to an army. According to the historian Zosimus, Alaric (the leader of the Visigoths in 398) rebelled in 395 because he was not satisfied with his responsibilities in the Roman army (which may or may not be true, but tells us that he was a Roman officer). We do not have precise information about the composition of the forces which he led afterwards (appartently, a mix of Goths that had been settled in 382, of auxiliaries who may have been Goths, or at least Germanic, and of runaways slaves; none of these groups evidently includes anything else than soldiers), Zosimus (book V) also tells us that his army numbered 40,000 men. As always, we do not know if such figures can be trusted; in any case, Ammianus Marcellinus wrote that the Alamans fielded 35,000 men in 357 (in what certainly was an army), so we can at least say that this number looked plausible from the point of view of a late Roman writer.

So a Visigoth under Alaric in 397 would have been living not as a member of a wandering group of people, but rather as a soldier. Even better, in 397, these soldiers had received official sanction: Alaric had received the title of magister militum in Illyricum thanks to the Eastern Roman Emperor, Arcadius, and his chief eunuch Eutropius. As far as I know, we have no straightforward indication about the financial consequences of this decision. But in any case, Alaric probably could draw upon local resources, order requisitions, access local forts, etc. His average follower, to answer your question, would therefore have been fed and sheltered by the local Roman authority (and, in a way, was the local Roman authority).

I recommend Liebeschuetz' “Alaric's Goths: Nation or Army?” on this specific issue (in Fifth-Century Gaul: A Crisis of Identity?). More generally, Halsall's works (among which Barbarian Migrations and the Roman West, 376-568) are quite enlightening. Peter Heather, who used to be in favour of the “marching people” approach in his earlier works, seemingly has adjusted his tack (see his contribution in the installment of the series Studies in Historical Archaeoethnology about Visigoths (The Visigoths from the Migration Period to the Seventh Century)). Walter Goffart's Barbarians and Romans has been superseded on some technical aspects, but his general argument holds true (speaking about the Goths: “it is more evocative of the great company of successive condottieri than of a phenomenon of popular migration”).