A woven basket can become a sort of accidental kitchen scrapbook when remnants of things kept in it -- seeds, crumbs, old flower petals -- get trapped in the weave. Linen-lined baskets, used in Europe to hold rising bread dough, offer a lovely, practical solution. Many stores carry them, but it's easy to make your own.
To measure a round basket, run a cloth tailor's tape down into it from lip to lip. (For a rectangular basket, repeat for crosswise measurement.) Adding 2 inches all around for a hem, cut a piece of unbleached linen to size. Center the linen in the basket. With heavy linen thread and a sturdy needle, sew the lining in place along the bottom inside edge of the basket using 1/2-inch stitches. (If the basket is tightly woven, use pliers to pull the needle through.) Smooth cloth, and pleat it to fit, making four evenly spaced folds (at corners for a rectangle); secure each fold with one big stitch the depth of the basket. Stitch linen around the basket's outside lip, turning the hem under as you go
.

Linen is or was an extreme pain to iron......... who needs it? I would use cotton calico any day and even a cotton blend for the showy stuff, other wise unbleached muslin.. And just hem the liner and make it so it can over hang the basket, after preshrinking muslin in HOT water and rinsing with a bit of vinegar.
Hi! This is actually my "traditional" baby shower gift. I use bright fabric, a large basket, stitch so that it can be taken out and washed. My daughter will use her's for the baby's christening basket...BEAUTIFUL!!
I made a few of these liners, but I ran a grosgrain ribbon through a casing so they would be washable. Also the bow looks festive.
How about sewing an elastic band around the lip of the linen insert? Easier to clean, no ribbon to add/remove to clean, easy to clean.
Sew the linning to the basket? How do you take it out to wash it ????
If you made an additional "napkin" out of the same fabric, you could use it along with the basket to cover up warm rolls or bread at the table.
I know it might knock down the esthetics a bit, but I think I would use a similar-looking fabric, but one that washes up with less wrinkles. I also like the idea of elastic or a drawstring and ties to keep it attached to the basket, but have easy removal, too.
. . .
I have to agree with pioneer588, these look a little half done stitched to the basket! I would probably add extra seam allowances for a double fold hem and use a bodkin to pull some nice cord/rope/ribbon through it so I could easily tie it on
Where would I find unbleached linen and linen thread. I have some linen and flax fabric. Would that work? Should it be laundered first, especially if it is going to be used for raising dough?
Elasticating the hem is the better suggestion.
The lined baskets look so beautiful and I'd never thought to rise dough in them. Instead of sewing the lining to the basket I think I'll run elastic through a hem casing so I can wash the lining easily.
This would make a nice gift also by using pretty dish towels or napkins to line th e baskets, depending upon the basket size.