I would put something, even with just cloth there will eventually be a bit of wear on the wood from pulling it back and forth, and poly is harder than the wood and will protect it from that. Also, the curtains will slide much easier. Also, a finish slows the rate of moisture movement into/out of the wood in response to humidity changes, which reduces internal stresses on the wood, which might lead to warping. I doubt that would be much of an issue with pieces of wood this size, but you never know.
Poly will help keep the stain from leaching out into the curtain fabric.
I would put something, even with just cloth there will eventually be a bit of wear on the wood from pulling it back and forth, and poly is harder than the wood and will protect it from that. Also, the curtains will slide much easier. Also, a finish slows the rate of moisture movement into/out of the wood in response to humidity changes, which reduces internal stresses on the wood, which might lead to warping. I doubt that would be much of an issue with pieces of wood this size, but you never know.