Can a lack of sleep degrade balance?
Short answer
Sleep negatively impacts attention, which in turn impairs balance control in cognitively challenging situations.

Background
Schlesinger et al. (1998) argue that under normal conditions, postural control appears to be automatic, and to require little or no attention in young, healthy adults during quiet standing with full sensory input. However, postural control appears to require increased attention when sensory integration or motor coordination are required. Examples of such challenging conditions where increased attention is required are situations during reduced or conflicting sensory input conditions, or when one is subjected to balance perturbations. The regulation of attention, in turn, is dependent on sleep, because sleep deprivation causes deficits in attention. Hence, although sleep deprivation per se has little or no effect on standing postural control, it can impair postural balance when one has to perform a cognitive challenging task. 

Reference
- Schlesinger et al., NeuroReport (1998); 9(1): 49-52