Ethernet Cable Through Floor
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.