If you have a landing, I would suggest you start on this mitered seam (angled cut edge of the rug) as the first point of installation. Then work your way up and down from this seam with the 2 joined runners.
Depending on the space in your stairwell, you may want to create this seam on the floor in a more open space. If you decide to do this in place, be sure to mark the placement of the runners, with painters tape to keep things square and straight as you install.
To determine the mitered seam, lay out the rugs face up in their proper place perpendicular to each other with a full square of overlap. (The shape will look like an L and the corner of the L will have two layers of runner overlapping.)
Fold back the corners that are overlapping, and create the 45 degree fold down the center, like the diagram shows. You will be seeing 2 triangles of the back side of the runners, when you fold them back.
These triangles will need to be trimmed off. How much you trim is determined by the thickness of the carpet itself or how much the runner will unravel after being cut.
If you have a thick carpet you will not leave any excess but cut it exactly together along the 45 degree angle and use carpet joining seam tape to tape the carpets butted tight together at the seam.
If you have a thinner runner, then I would leave an excess of 2 inches past the folded edge. Cut off the triangle of carpet that will not stay leaving 2 extra inches like shown in the diagram below. Fold the 2 inch piece back onto itself under the face of the runner. Secure that cut edge, by adding the double sided tape and folding the excess back edge to back edge (with the tape sandwiched between).
This will give you two finished 45 degree angle cuts. Now you can use carpet seam tape on the back edge to secure the two pieces together.
Once you have the two rugs in place on the stairs. square and ready, use the double sided carpet tape on the edges of the rug to secure this piece to the wood. Then staple along all edges and get the runner nice and secure. If necessary you could add some staples to the mitered seam, but these may be very visible or felt by bare feet, so be careful.
To finish just work your way down the stairs or up the stairs from this seam, following the carpet runner installation directions above.