Hi brinemobee, I just ran into the same problem while modeling a strand of DNA. My solution is to duplicate the curve and move it up (or down) along the axis. Then use alt-c to convert the curves to meshes, join both meshes (ctrl-j), select everything in edit mode and use bridge from the looptools add-on to fill in faces. Now you have a thin ribbon winding around the axis. You can extrude the faces with 0 displacement and then hit alt-s to extrude them inwards. Or use the “inset extrude” addon to extude them inwards.
If you don’t like the whole extruding idea, you could just make a second curve with a slightly smaller radius and duplicate that one along the axis, too. Use bridge or loft in the looptools add-on to skin everything once you’ve turned them to meshes.
I hope this is clear enough, otherwise let me know and I’ll post some pics.