Fixing broken old speaker

The speakers can deteriorate on their own in less than 35 years (i.e., it could something besides little fingers).

If the surround on the speaker has deteriorated, then you can purchase kits from Parts Express and other places than allow you to fix the surround yourself.

If it is a well-known speaker that sold a lot, a few places (Parts Express) can sell you a replacement speaker with the (approximately) same Thiele/Small parameters. You cannot drop in an arbitrary driver except when its T/S parameters are approximately the same.

My rule of thumb is that speaker drivers improve significantly every ten years (not always true). You may be better off asking individuals in this forum about what replacement speaker to purchase. If the speaker is a high-end speaker and the cross-over has not deteriorated (typically the electrolytic capacitor), then it may be worth it to pursue a new driver.