Since she seems to be accepting of the crate when she can be with her humans, try putting the crate wherever you are. At night, next to the bed so she can smell and hear you. As long as the crate is the right size, this should work. After a few days, move the crate to the other side of the room and eventually to the location where she will always sleep.

Also, keep the crate in whatever room you are both in during the day with the door left open. Toss in a treat occasionally so she can go in and get it without the door being closed behind her.

Right now she is feeling insecure. If you can be in the same room as she is and keep an eye on her, don't crate her. Otherwise, be in the same room with her in the crate and talk to her as much as you can. I would avoid putting her in the basement alone. It is a good place to play with her, but not to be banished to.

I hope this will help. Rescues are a whole different thing than new puppies!