The best kind of wood for interior wooden shutters is Basswood. There are many reasons for this but here are just a few:
1) Basswood is a light wood but it is also a strong wood
2) It has a grain that is uniform allowing a beautiful stain finish
3) Basswood is easy to glue and to finish
4) Basswood trees are growing faster than they are being harvested
5) It does not warp
The last item in that list is a very important one. Basswood does not warp and that makes them great for being next to windows where they will be exposed to sun, cooler temperatures, and some moisture. Wooden shutters can be a problem if a wood is used that does not weather the conditions properly.
Basswood is a wood that is easy to work with because it glues well and can be made into the texture you desire by sanding and staining. It also dries rapidly without any distortion. These are all very important to have in a high quality wooden shutter.
Basswood is also often used for such things as furniture, wood handles, millwork, musical instruments, and of course interior wood shutters. It is a wood that is useful to make many different things and if you are contemplating getting wooden shutters, basswood should be at the top of your list of woods to consider.