If keeping the basket maybe add hay or a paper lining or cardboard floor if there is the risk that little feet get stuck in the weave. Looks like there might be the risk of tipping if the doe steps on the rim (stuffing hay around on the outside could fix that if it's an issue), and it looks like it will be too easy for kits to get out or get draged out on the teat, which can become a problem.
If using a nestbox just move the whole nest, hay, fur, kits, into the nestbox, you can put some hay as bedding in there first. Place the nestbox where the basket was. Be sure the kits are well covered with fur.
When I had to move nests to a different location (from a tunnel in the garden to the hutch) I showed it to the does by stuffing her nose into it, limiting her space around the nest for a day, and leaving her alone. But that shouldn't be an issue if it stays in the same spot.
Does go to the nest about twice per day, for just a few minutes to nurse, Some does do not go near the nest when someone is around (to not lead predators to the nest), so give her some privacy, especially around dusk and dawn. When there is no problem, don't seperate them, you can lock the doe away when your'e checking or cleaning around the nest though, shouldn't be a problem.
Let her do her thing, don't meddle too much - that little voice in their heads, their instincts, can be droned out too easily. Seems she is doing a good job.
I initially check the nest for soiled stuff or dead kits, then leave them pretty much alone for the first week or so, just putting the hand on it once a day to feel if everything is ok.
Feed the doe well, you can tripple her pellets, and give her unlimited hay. Never had a nursing doe get fat, they need a lot of energy.
