Basketweave stitch is made of blocks of knit and purl stitches alternating both across the fabric and along its length. For worsted weight yarn 3 stitch by 4 row blocks of each are a good choice. Knit 3, purl 3 across every row for 4 rows. Then purl 3, knit 3 across for 4 rows. On that 5th row, all columns of stitches that were formerly knits change to purls and vice versa.

Other stitch and row counts may be used. There should always be more rows than stitches per block because rows take up a bit less length than stitches do width. 4 stitch by 6 row blocks usually look good.

The sample scarf is made with #4 yarn, specifically  Premier Everyday Worsted, one of my favorites.  The knit-as-you-go trim included in the description is a form of full fisherman's rib.

SCARF SPECIFICS
32 stitches will be about 7" wide. You may widen the center section by adding stitches in increments of 3 to make wider scarves, blankets and shawls. You may use cotton yarn and fewer rows to make dish cloths and pot holders. Basketweave may be knitted using either standard purling technique or the no-purl double board method. The no-purl method is described in the instructions  below.

1. Cast on using figure 8 across a span of 32 pegs, alternating using a peg from one board, skipping a peg on that board and next wrapping a peg on the opposing board. Leave a 36" yarn tail before knotting a loop into the cast on yarn and beginning the figure 8. This sets up for knit 1 purl 1 ribbing using the no-purl method.
2. Rearrange stitches so that, beginning at the left, the first 4 alternate 1 on the near board, 1 on the far board. The next 24 alternate 3 on the near board with 3 on the far board. The final 4 alternate 1 on the near board, 1 on the far board.
3. Always slip the first stitch of the row. Slip means, neither wrap nor knit the peg, when loom knitting.
4. On all right to left rows, that is, rows 1,3,5,7 of the 8 row repeat, slip peg 1, wrap but don't knit pegs 2, 3, 4. Wrap and knit the 24 center stitches wherever they are. Wrap and don't knit pegs 4, 3, & 2 at the left end. Wrap and knit the end peg.
5. On all left to right rows, that is, rows 2,4,6,8 of the 8 row repeat, slip peg 1. Wrap pegs 2,3, & 4.  Also wrap and knit over the first peg in the basketweave section to anchor the yarn. Knit 2 loops over 1, first on peg 3, then 1, then 4. Work the 24 center basketweave pegs wherever the stitches are. Wrap pegs 1-4 from the other end. Knit over peg 1, the end peg. Knit 2 loops over 1 first on peg 3, then 1, then 4.
6. After completing row 4, move each stitch of the center 24 to the opposite board from its present location. Work rows 5-8 as rows 1-4 were worked. The description is the same but stitches that were formerly being knitted are row being pulled and purls are being knitted. After row 8, move the center 24 stitches across to the other board again. Repeat the entire 8 row sequence.
7. Work to within 2" of the desired length. Change stitches back to the ribbing setup. Knit 12 rows of ribbing. Bind off in any neat way, leaving a 36" yarn tail. Use the yarn tails to chain stitch across the ends so that they match. 

The Loom used for the sample is a modified 28" KB with 7/16" peg spacing. Details are in this video.