Nothing much happens. You get Valens around a little longer. Eventually some other "barbarian" tribe either gets settled within the frontier as federates, as the Franks had already been, or is allowed to enlist in the Roman Army en masse as a tribe, and not necessarily as a result of a defeat. The arrangement with the Franks was made after the Romans beat the Franks in a big battle near Strasbourg.
You still get civil wars, and its only a matter of time before some general thinks its expedient to make an alliance with one of the barbarian tribes.
There are some effects as not having the Goths around as a contender, but then they could come back. The Ostrogoths spent the century as lackeys of the Huns, and then seized Italy. The Lombards later came pretty much out of nowhere. Someone else, likely the Vandals or the Gepids, will grab Span if the Visigoths don't.
The big break, the crossing of the Rhine in 406, came after troops were taken from there to be used against Italy. They happened to be used against Alaric, but any group of out of control federates or a civil war contender will do.
The problems with the Roman Empire at that stage, namely constant wars and plots around the Emperor, a long term demographic decline, a breakdown of the machinery to recruit natives into the army, and a series of barely competent Western Emperors, don't go away just because the Goths do.