$20 for a studfinder is the best investment you'll ever make. More expensive ones can tell you if there's electrical behind there, less expensive ones are simply frustrating and a waste of money.
Sounds like you're JUST on the other side of the stud. Get or rent a right-angle drill attachment for your drill. This should fit inside a hole big enough to allow an electrical box.
Pull the cable up and through the hole in the wall. Knock out one of the plugs in your electrical box (shouldn't need more than a single-gang box, and those knockout plugs are just slugs stuck there for just this purpose) and pull the wire through your knockout hole.
Attach the electrical box to the stud, and cover it with a standard wall plate.
The only thing throwing a wrench in this plan is if you're a lot further from the stud than a single-gang box. You could go with a double/triple gang box if you don't mind the extra room.
How close is this to the fridge? If you feel like you MUST drill into the floor as your last resort, behind the fridge is always a nice place. Trim the molding behind there and throw a surface channel there to run the wire out from behind there and to protect it from being smashed by the fridge.
If it's not close to the fridge, getting more involved will give you good results - get a little prybar (DON'T use a screwdriver, you'll end up honking up the trim that way) and remove the base trim. Drill as close to the wall as possible from UPSTAIRS once you're sure there's no electrical you'll hit. Pull the wire up through there. Once you've got the wire up there, then you can place your outlet box somewhere. When you drill the hole for that, put a hole into the wall right where your wire comes from the basement. It'll be covered with the base trim when you put it back on. Drop the snake down, pull the wire, fasten everything back together, and you're done.