Can mobs spawn on ladders in 1x1 shafts?

I have a few access shafts that are 1x1 with ladders along one side.

The areas at the tops and bottoms of these shafts are well-lit, but the shafts themselves are pitch-black most of the way.

Do I need to light these shafts to prevent mobs from spawning in them?

According to the wiki, mobs can spawn in ladder blocks, along with torch blocks. You will need lights somewhere near the ladder to keep the light level high enough to avoid it from being a valid spawn location. This only applies for if the ladder is on the ground though, as mobs cannot spawn in the air. The block they spawn on must be opaque, or otherwise have a solid surface to stand on. The wiki page goes into full detail about what creatures can spawn where.

In your scenario, as long as the area at the base of the ladder is lit, and there are no empty pitch-black areas in the shaft leading to the top and bottom of your ladder shaft, mobs should not be able to spawn because of the lack of solid unlit surfaces for them to stand on.