What spoils breadcrumbs quicker, oxygen or moisture?
Oxygen will only spoil breadcrumbs if the bread was high in fat. The fat will oxidize and go rancid. Fat will also go rancid in the presence of light so a plastic bag on the counter would not be ideal either.

Moisture will spoil any bread immediately because it will allow bacterial and mold action. Where oxygen might make high fat bread go a little off after a week, moisture will make it mold in a day or two.

Just toast your breadcrumbs to drive out excess moisture before packaging in a plastic bag. If it is high fat bread then you might also keep it in the freezer.