For those who need a bit more help visualizing the woven basket stitch, there is a tutorial available on youtube here.
Note: It is also possible to knit this pattern as a 1×1 cable, using a cable needle (though it is much slower this way):
Row 1 (Right Side Row):
Pass the right-hand needle around the back of the first stitch.
Knit the second stitch on the left-hand needle through the front leg.
Without dropping the stitch just worked off the needle, knit the first stitch through the front leg.
Slip both stiches off the needle.
Repeat across row.
Row 2 (Wrong Side Row):
Purl the row’s first stitch.
Purl the second stich on the needle (through the front leg).
Without dropping the stitch just worked off the needle, purl the first st,
Slip both stitches off the needle.
Repeat across row until 1 stitch remains on the left needle, purl this stitch.
Repeat Rows 1 and 2.