All older eMacs have the same issues. They are old!
If you are running 10.2 (circa 2002) their best security days have long passed and cannot be regained with any update. The last for 10.2 can be obtained below but it was current in 2003.
The business model is such that the old are abandoned without care, to be replaced with the latest shiny new thing. Just hope Apple never get in charge of healthcare!
If you want internet security for banking or such, you will need to update the computer itself to something newer. Any Mac from 2006 which is an Intel, capable of running a Mac OS 10.6 Snow Leopard ($20) will currently afford you security.
Old Macs are still useful running their old software within themselves. I have an old iBook that will run Classic. Know the limitations though. Old computers are not up to scratch in a newer internet environment that is constantly evolving.
Software is fluid and changes, unlike old hardware which is fixed in firmware and time.