Check which port on the switch the old computer is plugged into. If it is labelled as an uplink/downlink port, try switching it to a numbered port, instead. Also, try plugging it into the same port that the new computer works on.
On some network devices, an uplink/downlink port can only be used to daisy-chain network devices to. It's a long shot, but might be the problem. Also, ports can occasionally go bad; if you try it on the known good one, you will at least know whether it's the port or the computer.
EDIT:
After looking at the additional details you posted and the documentation, here is the problem:
Even though the modem supports up to 63 users, it only does so if the cable provider does (see http://broadband.motorola.com/consumers/products/SB5101/downloads/SB5101_UG_EN.pdf page 40: Not all service providers support multiple user service. For information about multiple
user service, contact your cable service provider.)
And the switch documentation ftp://ftp.dlink.com/Switch/des1005E/Manual/des1005E_manual_110.zip page 15: 1. Why can´t I share my Internet connection to multiple computers when using my D-Link DES-1005E switch?
If you are connecting directly to a Cable or DSL modem, you will need a router to “share” your Internet connection or a computer using ICS (Internet Connection Sharing) or Proxy. Switches and hubs do not allow you to share a single IP address to multiple computers. Another alternative is to contact your ISP and purchase extra IP addresses for each additional computer.
So, here are some options:
1. Replace the switch with a router. This will allow you to use NAT to connect multiple devices, using the single IP address from your ISP for the router itself.
2. Set up the new computer for ICS, which the old computer can then access through the switch.
3. Get another IP address from your ISP for the old computer.
4. Only use one computer at a time; turn the other one off so the active one can get an IP address from the ISP
5. Replace the modem with a combination modem/router. When you initially reported that both computers were getting their own IP addresses, this is what I thought you had and it will work with your switch. However, after seeing the documentation for your modem, I realized the IP address that the old computer is pulling is probably the default APIPA address Windows assigns when a DHCP server is not available (169.254.X.X).
Personally, I recommend #1. It will allow you the most flexibility to expand in the future, if you need to (I have about 10 different devices connected in through 2 routers and a hub on my one cable line).