I "see" several problems here,but all are easily solved by using the right materials and procedures.
1.) DON'T use balsa for this type work, especially where you are going to stain rather than paint. Balsa is soft, easy to work with and readily available but it is better left to the model airplane guys. It warps very easily. should never get near any water. (we use a lot of water in building our layouts!), should not be assembled with white or yellow glue. Use "hardwood" instead, usually sold right next to the balsa, or available inexpensively from MRR scratch building suppliers in scale sizes. (I even sometimes buy some of the old kits that were started or are missing parts that still contain a lot of scale stripwood, just for the wood). A free source of boards like you used are the thin wood coffee stirrers available at many places where you buy coffee. If they are too wide for your project, you can cut them down with a new single edge razor blade or hobby knife guided by a metal straight edge. Making a simple jig is worth the couple extra minutes if you need to split a quantity. If you need boards exactly 1/2 the stirrer width, rather than split them, just draw a line down the middle with a sharp hard lead (H) pencil.
2.) DON'T use water based stains or paints on the raw wood. Check the stain label, if it says to thin or clean up with water, put it back. You can make a very good stain ( and very useful dark wash to bring out textures and variations in all kinds of modeling materials) that is quick drying and will cause minimal warping by adding a tablespoon or two of black india ink (NOT the new acrylic ones) in a pint of isopropyl alcohol from the drug store. You can make several color variations by adding a couple drops of brown and/or green India ink to a portion of the black wash. A quick way to use this for your unassembled stripwood is the put the cut to size wood in a ziplock freezer weight bag, add the wash to the bag and slosh around for a couple minutes. Remove the pieces and lay on some paper towels to dry for about 1/2 hr. Check the color after dry. You can re-stain some or all to darken more or give a sloshing in plain alcohol to remove some color. After fully dry (no more alcohol odor) you are ready to assemble. If you are planning to add a clear top coat coat like dullcoat or pledge floor finish, test this on the dry wood as it will darken the color.
3.) DON'T use white or yellow alaphatic resin glue (Elmers or Titebond) to assemble unsealed wood. They contain water which can also warp the wood. Don't use Gorrilla glue as it foams and expands as it cures. I suggest using a medium CA glue, used sparingly so it doesn't ooze onto the surface.
4.) If you are going to paint the wood with acrylic paints, seal both sides and all edges with a light coat of rattlecan automotive primer before assembly.
5.) If after all precautions, the wood still warps, don't despair, simply give the wood a couple spritzes of water and clamp or weight it flat until completely dry, at least 24 hrs.
I had thought glue was dried but perhaps not. My flooring did the same as well and I had to glue cross braces and put weights on to get it to lay down so perhaps the water based stain is culprit as MichaelE suggested..
