Why didn't the Byzantine come to the aid of the Western Roman Empire?

Although the Byzantine empire (formerly the Eastern Roman empire), made attempts, there were too many hinderances. They couldn't handle the challenge of the great distances they had to travel and they just could not respond to the scale that was necessary.

In the fourth and fifthe centuries, there was a great population drain on the Byzantine empire which made it difficult to survive, much less to help the Western Roman empire. They did not have the capability to produce the extra soldiers and weaponry necessary to fight wars in addition to protecting their own. Thus the Byzantine Empire tried to help, but they just couldn't handle the situation.

The Western Roman Empire was already weak for two centuries and helping them would really be a challenge for the Byzantine Empire. It would take the time to produce more soldiers, weapons, and resources to give aid to the western empire. By doing that, they might sacrifice a thing or two that can cause a downfall to their empire itself.

The safety of the eastern empire only mattered to them thus securing other parts of the Roman Empire was not of their top priority. All of their wealth was located in the East so they focused on protecting it as once enemies have invaded the East, their downfall will start and their empire will be out of their hands.