Crown moulding yes. Inside the beams - no. I have seen moulding on beams and it looks good only if the moulding is the same finish as the beam, and IMO you don't need it on the beams.

HI Jillian, thanks for your comment...I should have clarified...the picture I posted doesn't show the beams with the drywall etc on them....they look like wood beams, but in this pic they are just the metal beams prior to being drywalled....so they are now boxed and drywalled

Can you post some current pictures? Having a hard time visualizing your question.

Can you just do it on the back wall to trim the bookcases?

I think you could go either way since you drywalled the beams, though I don't think it's necessary. I'm leaning towards leaving it off.

I added beams to my den, which were made out of wood and painted white like our trim. We added crown molding all around the beams and the walls to tie everything together with the rest of the house. It created a cohesive look and it looked like the beams were original to the house. Point is, if the beams are painted, I would add the molding.

if you are looking for a cleaner, open look leave it off. you can always put it back on down the road if it feels too stark.

I finished my beams with plain panel molding, simply boxed them and painted them the same color as the the trim. My ceiling is 50% of my wall color so their is more cohesiveness to the room. Very pleased.

Even with the beams, you could still add back the molding painted the same white or off white color as the ceiling, giving the room a much more finished look without crowding/shrinking the room with too much dark wood and still have the beams you've invested in provide the solid look you clearly want for the room.

I prefer crown molding instead of leaving it plain.

It looks like you have a textured ceiling, putting crown could give you uneven look along the ceiling plane. In cases like that, we would mark the crown projection distance on the ceiling, snap lines and smooth the projection area flat so the molding will sits tight against the ceiling.

Either or, which ever you prefer but I think you would need to do the beams too.