The most effective way I have finished a woven strap was to take a piece of suede or any thin leather. I shape it sort of like an elongated football, doubling it to make sure the rounded ends are matched. I use a tacky glue to fold the leather over the end of the strap for about an inch. It can be straight and the width of the strap, once in a while I like to put a bulge in for aesthetics, whatever... If the ends are loose or frayed, some Bordens will seal them.
Sometimes I'd work in a D ring, sometimes I work it so that the fold of the leather is about a 1/4 inch longer than the end of the strap for a lace to pass through, sometimes I put on a buckle...it's up to your taste at the moment.
Then I use an awl...NOT a hole puncher that will cut into the fabric... and punch out a stitching design such as a weeping heart, or a flat seam that runs along the edge of the leather...whatever
Per above, a buckle is the best way to provide for adjustment, but the guys have provided some good advice in this thread otherwise.