I have a sink in my basement which I would like to permanently remove. As far as plumbing goes, what would I need to do to permanently shut off the water supply?

I've only done basic plumbing jobs, so I'm hoping the experts on here can help me with suggestions.

I would guess that I would have to cut the drywall around the water pipes so I could cut the pipes adequately so they don't stick out of the wall. Then I would install shut off valves and leave it permanently closed. Then I would simply add drywall to cover the hole I made and patch it up. Am I on the right track?

Very close but no 100% yet. You may need to open the drywall to get inside to cut some pipes but shut offs should not be left permanently inisde a wall. Cap the supply pipe with a threaded cap if galvanized or a sweat cap if copper. You should remove the trap from the drain and the lateral to the drain stack. That too should be capped.

Thanks for the reply ballengerb1. I have copper lines so I would solder on a sweat cap. Wouldn't water still try to force itself through the supply pipe?

Sure it will and at about 45 psi but your soldered cap will hold that 24/7/365. Can't say that forever with a shut off valve. I am guessing your trunk line is 3/4" and the taps to supply the sink are 1/2". Cut the 1/2" pipe about 1" out from the connection to the trunk and solder a cap there. Easier and cheaper than trying to remove the Tee and splicing in a coupling.