On paper, the western roman empire is far more defensible. It has a series of natural and easily defensible borders-the Alps, the Pyrenees, the Rhine, the Danube. The east, by contrast, does not. They have the Danube, but on the eastern frontier it's just open plains (of course that was rectified by massive fortifications everywhere). What the east had in the 5th century was a strong civil administration essentially controlling the empire (rather than the military administration, as in the west), and lots and lots of money (and grain). There's a good book, "The Grand Strategy of the Byzantine Empire" that goes in depth on this, but basically the east was able to play their enemies off each other and pay anyone off when necessary. That was their strategy well into the middle ages, and it worked wonders. They also of course had the one major key defensible border, and that was the Propontis straits separating the Balkans from Asia, which came in handy to save them on more than one occasion.
So the key to keeping the west alive is, first, keeping Spain and North Africa untouched. Spain (quite literally) was one giant goldmine for the empire, and Africa was also quite wealthy and of course the empire's breadbasket. The west's finances will be in far better shape if these areas remain untouched. So that means keeping the fighting contained to northern and central Gaul, north Italy, and Pannonia. There's a relatively easy way to do this actually. Just delay Constantine III's crossing into Gaul-or better yet, have him killed by the British soldiers too as his immediate 2 predecessors were, and have the British legions continue to squabble among themselves. This means he does not cross into Gaul while the barbarians have crossed the Rhine and allow them into Spain. Stilicho, by extension, does not get offed and is able to deal with the invaders relatively easily (this shouldn't be a problem really. The Franks very nearly defeated them at the Rhine crossing, so they would presumably be weakened and be easy prey. Alternatively, this also means the Romans could peel them off piecemeal with bribes to fight for them, as Stilicho did with Radagaisus). Even if Stilicho is offed, they will likely contain themselves to raid Gaul, still making them easy to deal with.
There's no real reason why the barbarians should ever really be able to reach Spain (and even less chance of them getting to North Africa) without a set of extraordinary circumstances. Now, the second thing here is the Goths. Even if Stilicho goes, keeping the Goths out of southern Gaul should not be a difficult prospect. Not much has to change to prevent Alaric from sacking Rome, and then as Stilicho showed, even if has to come to a costly battle, the Roman armies are still strong enough to force Alaric back into the balkans, where hopefully the Goths will be contained (this is where greater cooperation with the east is necessary, and probably where replacing Stilicho would be helpful given his combative relationship with the east).
Now after Stilicho, have Constantius still be raised to the purple by Honorius at some point, and just not die. He'd do a good job at stabilizing what's left to need to be stabilized and, more importantly, he'll re-set a precedent for powerful general aspiring to be emperor themselves, rather than just trying to control a weak emperor. Which is, imo, an important distinction. Then there's more than enough capable late western empire generals from OTL who could fit the bill of competent emperors to guide Rome through the 5th century crisis. After that, you can't really predict anything, since all the butterflies, but you could craft a world where the west survives.