@KatyBear Welcome to NGA! Boston ferns are fairly resilient when it comes to bumping them up to a larger pot, or even to dividing them. I have been known to use a sharp knife to divide a fern and it doesn't hurt them a bit. They just keep on growing.
Without seeing a photo of the fern in the basket, I'm going to guess that some of the fronds coming out of the bottom of the pot may have to be sacrificed in the process. That said, you may be able to, instead, cut the bottom of the hanging basket to get the fern out. Either way, your fern will be fine.