How do I get a turkey baster out of a carboy?

By this time you will probably need to let it sit in there. If it's floating near the top, I've gotten things out of carboys with chop sticks before.

The best way to get the turkey baster out with the least consequences on your wort is to wait until the beer is finished fermenting, and then just dump it out after the beer has been racked away. Whatever contamination was going to happen has already happened (hopefully you sanitized the turkey baster). Trying to fish out the turkey baster is going to be challenging, and likely to introduce oxygen or unwanted bugs to the carboy. This will require quite a bit of patience, or RDWHAHB zen.

My kid dropped a thermometer in an open bucket full of wort (pre-aeration), so I figured I was safe to fish it out with a couple sanitized spoons. But it was impossible to get it out. I eventually had to fish it out manually, wearing a long and sanitized rubber glove. It's not as easy as you would expect to fish a long, thin thing out of a turbid liquid. I can't imagine trying to do it through the neck of a carboy.

Depending on how wide the small end of the baster is, I'd consider using a wire coat-hanger, with a very slight hook on the end. Sterilize it by curling it up, and boiling it in a sauce pan full of water for a couple of minutes, using a star-san soaked rag to grab hold and straighten it to fish around for it.

On the other hand, perhaps just leave it until you syphon off the beer? If it had contaminants on it, they've already been introduced to the solution, no need to risk adding more unless you really need that baster.