When our EIGRP routing was first setup the installers didn't secure it. We now have 25 routers, and it is way past time to secure it. I have read and understand how to enable and have done so on there networks with OSPF, but not once the network is using it.
How can I set this up without taking the network down?
Once MD5 auth is enabled on all interfaces then changing the key can be done without any downtime.
But as Jerry notes this won't work when you actually configure it for the first time.
If no downtime is acceptable then one way around this is to configure a second EIGRP AS on each router. Because authentication uses the AS number a second AS would allow you to break the original AS while adding MD5 auth and the router will still have all the routes in the second temporary AS.
Once MD5 is enabled and working in the original AS then you can delete the second temporary AS.