Fringe and Hemstitching
I have now started my first “proper” bit of weaving and have completed the first of a series of table napkins. I used the hemstitch from The Weaver's Companion but I am not sure if this is stable enough unless I allow for fringing. I do not really want them fringed, so would it be better for me to fold in the ends and machine stitch them? Would it be better if I wove a half inch of plain weave at the beginning and end and just machine-stitched them? Either way how much space should I leave between each piece?
There are several choices to make with hemming and/or hemstitching. First of all, because you would really like napkins to endure repeated washings, fringe is not a good idea (as I think you have surmised). A finish of hemstitching with a trimmed fringe would not hold (no matter what hemstitching method you use). You could turn the trimmed hemstitched end under once (to avoid the bulk of a double fold) and sew that hem by hand or machine, but I think it would still have some weakness and also would not look appealing on the wrong side.
For napkins, I would weave about 1/2–3/4 inches of plain weave on each end (1/2 inch for fine threads, 3/4 inch for heavier ones) and separate napkins during weaving with 1 or 2 picks of a contrasting color. Also, begin and end the entire length of fabric (with all six or eight or however many napkins in a row) with a few picks of extra plain weave. When you remove the fabric from the loom, machine zigzag in those extra rows to secure the two raw edges. Machine wash the fabric in the same way you plan to care for the napkins (probably hot water, regular cycle, mild detergent). Machine dry until partly damp and press. Then, cut the napkins apart as you are ready to hem each one, turn under a narrow hem at each end, and sew hems by hand or machine. Some weavers think all handwoven linens should be hand-hemmed, but I prefer machine stitching for napkins for greater durability.
And just as an aside, when hemstitching is your choice, double or Italian hemstitching provides a more secure fringe and a more decorative look than single hemstitching (see Weaving Well for a demo and this free download for instructions).