Do you have any land line phone service now? Or, do you have phone service as part of your cable subscription? In either case, these wires may still be in use. If your answer to the first question is "no" and the second is "yes", go to your cable modem and look for phone wires connected to it. I'll give you more suggestions depending on your answers to the above.

Thanks for the response. We do not use a land line phone and have no plans to. A basic phone service is part of our cable/internet bundle, but again, it's not in use. Attached is a picture of our wireless router (in a different room). Thanks again.

Have you looked to see if you can locate the other end of the cable? Maybe someplace in the basement or maybe another phone jack?

It's an old apartment building in NYC, and it doesn't seem to end anywhere. It just goes into the wall as pictured. Can't find another phone jack anywhere else in the apartment.

OK - assuming that you don't own the building the 'proper' thing to do is probably buy a new surface mount telephone jack and leave the wire so that it could be used by a future tenant.

If you don't care just cut the wire off and shove the end into the wall!

Thanks! And if I take the "shove into the wall" route, do I need to be concerned about the wires from an electrical standpoint?

Well - if "you don't care" then don't be "concerned". :-)

It's not like you are going to burn down the building or knock out all the power to NYC.

I appreciate Corinne's concern. Is the cable shown in your first and third picture loose? I.e. could it be pushed back into the hole in the baseboard? If so, I would remove the wires from the terminal box (by loosening the screws), bending them back, and wrap tape around them, then pushing them all back into the hole. If you can't find other jacks, then the other end of the wires are probably in the basement. Can you ask the super?

If it's a rented apartment you should be doing anything because it's not yours. If you want to improve the appearance, a new phone jack is about $5 at the home center. You can probably leave the wires and base alone and just use the new cover (a single screw usually holds it in place)

If you own the apartment and aren't worried about the next guy, you can just cut the cable clean across and shove it back in the hole. It's low voltage and doesn't need to be capped.

Ditto to all mike said.