Network transfer between the two computers will be slow, for sure. But your slow computer isn't going to make your other computer run slower, unless you start having it constantly access the fast computer over the network. All the slow computer can do to the fast one is send requests for hard drive accesses (iff you have shared folders) and make the fast computer respond to them.
However, you might have to be worried about the GoodTimes virus... from what I remember, it was released into the wild sometime around when that old computer was still being used, and it's completely undetectable and can never be eradicated. It's an airborne virus, but it can only be spread within a small radius so your fast PC is probably still clean, however it could spread to the fast computer over the network quite easily. Signs of infection include misplacing your car keys when you were sure you left them on the table, getting popups in Internet Explorer, dropping the soap in the shower, and sudden unexpected "talks" from your significant other about your relationship.