Tom,
I was guest 'curator' for a museum knife exhibit a couple of years ago. It was very helpful to me in terms of displaying knives. I agree with the Count...highlight the knife and keep the display simple. Determine how to best 'show' the knife. For example, would it look better angled tip down at 30 degrees or horizontal and canted towards the viewer? Think about how the knife is viewed at different angles as the museum guests will be of varying heights. Don't underestimate the importance of good lighting. We changed numerous displays because of available lighting. Are you able to choose your display? The museum should be able to help you with these questions.
As to construction of the displays, we used two methods. The first was to take assorted blocks of wood (square, rectangular, wedge) and mix and match to support the knife. Once you are happy, you can use wood glue or hot glue to assemble the pieces for the base. Add pins, if needed, to retain the position of the knife and cover the base with black felt (or a covering of your choice). The second method was to use a square or rectangular block for the base and pin stock to create an upright rest for the knife. I used a tubing bender to get nice curves and you can support the handle or guard in a number of ways with the knife either tip down or tip up (typically at an angle and on a bias). I used 1/8" pin stock for most everything, but occasionally I needed 3/16". I used clear shrink tubing to cover the part of the rest that came in contact with the knife.
Congrats on getting in the museum.
Brett