The issue, at least as far as using EOS Utility is concerned when using just about any current Operating System, is that the current version of EOS Utility no longer supports tethering to the original 5D.
If you choose to use EOS Utility, be sure to download it using the 5D support page, as the latest versions of EOS Utility no longer support the 5D. By using the 5D support page and designating the OS you are using on your device, you will get the last version that does support the 5D (ver 2.7.3). The last Mac OS that shows an EOS Utility download from the 5D support page is OS X (with no updates). I do not know whether EOS Utility 2.7.3 will work on a later version of OS X.
For the Windows folks, the latest OS that is supported for EOS Utility 2.7.3 is Vista, so you'll have to choose that to get the EOS Utility 2.7.3 download. Even though there is no version of EOS Utility compatible with the 5D when Windows 7 is chosen as the OS, it will probably work with Windows 7. (I've found this to be true with other older versions of Canon software that had no version listed for an OS released after the Camera was no longer supported by updates). And with versions of Windows newer than Vista you can always try running EOS Utility 2.7.3 in compatibility mode.
If you can find an older version of Lightroom or Aperture that supports tethering to the 5D, you will also probably need to find a workaround to run the older version on a newer OS. But since EOS Utility provides more functionality when tethering than Lightroom or Aperture do, I would start with EOS Utility.