We recommend that two Cat 5e (or Cat 6) cables should be run to each score bench from the internet connection (office router?). You never know if this is going to be useful for future use and generally it is cheaper to run two at one time rather than a second one later.
The cable should be terminated at a wall socket underneath the scoring bench. This means that the Stadium Scoring computer will need a fly lead (short lead) to connect it to the system. Doing it this way gives you more flexibility for future use.
Enclosure and Security
We would also recommend that you enclose these sockets (1 or 2 depending on how you go) in some kind of box. Essentially you want to be able to stop people from being able to unplug the cable (breaking the systems - and being very annoying to trouble shoot). The box should have holes to allow the cables and fly leads to come in and some kind of door to allow you access to repatch if required. The box should have some kind of keylock (or padlock) to allow you to secure it.