My basement is damp. Will using bamboo there be a problem?

If you use bamboo on the ceiling, it is probably not a concern. The walls may be more problematic. If you elect to go with the bamboo bricks, there is little or no problem in them getting wet. If you go with bamboo half-rounds individually installed, you should be concerned about the inside of the half-rounds geting moist and mildew and other bad stuff growing unseen there. I would recommend foaming the inside of the half rounds with a closed cell foam to preclude stuff filling the space and becoming an unseen problem.