Many people learned that the fall of the Western Roman Empire came about due to invading barbarians. There is an element of truth in this, but it would be closer to reality to say that these were immigrating barbarians.
For these groups were not setting out to conquer the Roman empire. For the most part what they wanted was to become a part of the Roman empire: to settle in it and to reap the advantages of its law and order and economic prosperity.* Alaric, who famously sacked Rome in 410 A.D., had held high positions in the Roman military, and was not invading so much as protesting maltreatment.
If Rome had adopted open borders, would this have fixed the problem? It seems doubtful: my tentative judgment is that the Western Empire simply would've been overwhelmed earlier: while the Romans were great assimilators, it took several generations for the process to work. If too many people came in too fast, Roman institutions would be overwhelmed before assimilation took place, as eventually transpired.
* One can tell a pre-Roman town in Gaul from a Roman-era town by looking at how and where it was built: The pre-Roman towns (oppidum) would be built on the top of hills, and would be heavily fortified. Once Gaul became a part of the Empire, the hill-towns were abandoned, and unfortified settlements taken up in the valleys. Consider the cost, prior to mechanized transport, of having one's town on the top of a hill: The rich farmland and river transportation were down in the valley. An enormous amount of time must have been spent going down from where it was safe to live to where the economic action was, and then back up again. The benefits of being inside the Empire would have been tremendous.