The reason for using a basket weave selvedge is to keep your edges neat and even and to avoid having to use floating selvedge threads. Their beauty is that as you throw the shuttle back and forth, your weft catches the edge thread of your woven piece on each throw. Another reason for using the basket weave selvedge, rather than just a plain weave selvedge (using only two shafts) is that plain weave selvedges take up differently than your main pattern and will give you a ruffled edge, whereas basketweave seems to truck along nicely at the same take up as the rest of your piece. One of the pitfalls in using this treatment on everything you weave is that it uses up four of your precious shafts.
The very first thing you need to think about when trying to add this selvedge treatment is that a basket weave selvedge not only needs four extra shafts, but four picks as well. The biggest mistake that I have made is thinking I have four shafts for my pattern and four shafts for the selvedge. But, if I am weaving with a point treadling my sequence will be 1-2-3-4-3-2 which has six picks. I can manage to weave this if I have a table loom and put my draft into a liftplan or I might be able to work out a more complicated treadling sequence with my floor loom, but it's going to be awkward. Everything needs to end up with a treadling sequence that is a multiple of four so in my example above, the minimum treadling sequence is going to be 12 (which is divisible by 6 - the pattern treadling and 4 the selvedge treadling).
Once you have decided that your pattern choice is going to work and you are going to proceed, it becomes more straight forward. I usually put my selvedge threads on the last four shafts, but when working with a multishaft loom, sometimes I put them on the front four shafts for easier repairs should a warp thread break. (This was a tip given by Allen Fannin on Weavetech many years ago).
The example below shows the threads on the last four shafts which have been colored blue in the warp. (Half the threads are on the right side and the other half are on the left side.) I have used five threads to make the floats smaller between the interior pattern and the selvedge threads, but you can just use four on each side if you like. Notice the tie up on the last four shafts. In this example the treadling is a repeat of 8, so the tie up for the basket weave is repeated twice. Also, the threading for this example makes it necessary that you start throwing your shuttle from the right side. You can see that if you started throwing from the left side, you wouldn't pick up the edge thread and nothing would work right after that! I always keep my selvedge threads in this threading configuration so that I always throw my shuttle on the first pick at the right side of the warp.
The best description of selvedge treatments (that I have found) is in the book Handloom Weaving Technology by Allen Fannin. It is on pages 264 & 265. The book is fairly technical, but does have other things that will interest handweavers. If you are interested, used copies are fairly inexpensive.
After this post was published, another weaver friend pointed out that Peggy Osterkamp discusses the same topic in her book Weaving and Drafting Your Own Cloth on pages 88 & 89. She refers to this selvedge treatment as "tape selvedges" and also 2/2 hopsack weave. Peggy's book is crammed with information for weavers and would make a good addition to your weaving library.