National Electrical Code 2014
Chapter 8 Communications Systems
Article 800 Communications Circuits
I. General
800.24 Mechanical Execution of Work. Communications circuits and equipment shall be installed in a neat and workmanlike manner. Cables installed exposed on the surface of ceilings and sidewalls shall be supported by the building structure in such a manner that the cable will not be damaged by normal building use. Such cables shall be secured by hardware, including straps, staples, cable ties, hangers, or similar fittings designed and installed so as not to damage the cable. The installation shall also conform to 300.4(D) and 300.11.
Chapter 3 Wiring Methods and Materials
Article 300 Wiring Methods
I. General Requirements
300.4 Protection Against Physical Damage.
(D) Cables and Raceways Parallel to Framing Members and Furring Strips. In both exposed and concealed locations, where a cable- or raceway-type wiring method is installed parallel to framing members, such as joists, rafters, or studs, or is installed parallel to furring strips, the cable or raceway shall be installed and supported so that the nearest outside surface of the cable or raceway is not less than 32 mm (11⁄4 in.) from the nearest edge of the framing member or furring strips where nails or screws are likely to penetrate. Where this distance cannot be maintained, the cable or raceway shall be protected from penetration by nails or screws by a steel plate, sleeve, or equivalent at least 1.6 mm (1⁄16 in.) thick.
300.11 Securing and Supporting.
Please see NEC for full text
So basically, you're going to want to protect and support the cables in a similar manner to any other cables. Curiously, the code didn't mention 300.4(A) Cables and Raceways Through Wood Members.. Though I'd likely still follow that section if I had to run communication cables through wood members.
If you run the cable through conduit, you'll have to follow the code for protection and support (fill requirements do not need to be met, per 800.110(B).
If you run through a drop ceiling, you'll have to install support wires for the cable, or support it in another approved manner. You can't support the cable using the ceiling grid, or support wires.
Here's a great document from Mike Holt that explains Article 800 (with images).
Introduction to Article 800 - Communication Circuits (PDF)
If you can run the Ethernet to the basement through the cable box, why not just do that? Both the co-ax and the Ethernet cables are low voltage. There's no problem running them together in the same box and/or conduit.
If you run the Ethernet cable exposed along the wall, it won't cause data transmission problems, but it won't be attractive and you'll have to secure the cable to the drywall or the concrete blocks. If this is all indoors, you're not going to have a problem with the cable jacket breaking down. Although you can obtain UV resistant and direct-bury Ethernet cable if there's going to be exterior exposure.
Whether it's interior or exterior, you could also consider using conduit.
Bear in mind that Ethernet requires a minimum 3" bend radius--don't kink the cable or bend it at a sharp angle around or within any corners or you'll have data transmission problems (or failure).
Best Answer
I don't know how much code applies to low-voltage wires. Since it's unlikely this will be inspected at any point in the near future, I'm not sure it much matters.
You can drop Ethernet or coax through a drilled hole like this. Mind you, it's not sightly, but it sounds like that it's not a concern. It would be trivial to remove, since you simply remove whatever ends you've attached and pull the wire back into the basement.
If you have access into the wall, the cleanest solution would be to drill through the top plate of the basement and the bottom plate of the wall, then cut a low voltage bracket in and fish the wire up.