I have a Mac upstairs that's connected to Earthlink. I had a big oredeal with Earthlink a few months ago in which they insisted that I had a bad modem and they made me "buy" a new one by signing up again... (What a racket.)

Anyway, the new modem did NOT help. They actually had to have Bell South (AT&T) check the line and they found that my connection at the CO was corroded!!

So now I have this old modem. Can I hook it up to a Mac mini downstairs and use it with my Sony HD TV? If the Mac upstairs is in sleep mode would the modem dowstairs see the other one on the line and not function?

You cannot use two modems at the same time. You probably can turn one off and then use the other one. Whether the computer is on or not is irrelevant.

Assuming you have only one account, why not get a router for your modem that connects to more than one computer? Maybe a wireless one....

Probably he best setup in your case is a combination modem wireless router. In my experience it is more reliable than two separate boxes. BellSouth gave me a free one a few years ago, if you switch accounts you might get the same deal.

if you have 2 active internet accounts, there are ways to link them to so you can use both to connect to the internet. usually these routers are more expensive or are linux based. For example, Zyxel makes them (like the zywall 35) or i think pfsense can do it.

but if you just have a bunch of modems and one account, you can't use them all on the same account. the best choice, as was mentioned earlier, is a wireless router.

I'd return the extra modem for a refund. Then get a wireless router.