Here’s a picture:

You could paint it Linen White, or the drawers a Khaki/tan color.

I would paint with the same color.

What a nice vintage bath! Historically your bath would have likely been all white to match the notion of sanitary. I know it's pretty boring for a lot of people though. I would do it all the same color, whatever you decide. What is under the painted finish on the door on the right side of the photo? It would be a total bear to strip and stain it, but maybe that's an idea, if it was stained and laquered before?

All the same color, and carefully remove the hardware and strip it back to its original finish (probably brass plating) The stripping will be a lot of work but it’s the earrings on a black dress.

Really great bathroom storage.

Would paint it all the same white as the door and door trim with the walls a different color ... except that, if the two open shelves are removable, you could paint the "wall" behind the shelves the same color as your wall.

If the old paint job isn't entirely smooth, then the new paint job will not be either. Also, built up paint can make drawers difficult to open (as can being in a damp room) and white paint over a color can look different than white paint over white ... so I agree with HALLETT & Co. that it would be far better to strip off the old paint before repainting if at all possible.

This is really helpful. Thank you!

Sure white makes sense but I would love to see some color. The entire built-in, one color- could be neutral or a subtle color. Add beautiful hardware & vintage patterned wallpaper/fabric behind the open shelves. You can attach it to cardboard so as not to ruin your wood. GL :)

If the original hardware was brass, do I brass knobs on the drawers? What about fixtures?

Color Zen said exactly what I was thinking! A lovely subtle color and wallpaper behind the shelves. You can simply add wallpaper to foamboard cut to fit between the shelves. What colors do you have in the rest of the house?

There ia an issue with the wall in that area of shelves it will need to be addressed. As for the cabinets if it was mine they would be gone for some really useablee storage but since you love them all one color. I agree the hinges need stripping and that green trim above the door needs to be painted too. As for the color I think white would be original , love the glass knobs and amybe replace those shelves with glass and add some pretty soap and lotion bottles in colored glass .

I would paint them all the same color and would probably go with white. The darker the color, the smaller the room will feel. If you painted them all white, you could still add a pop of color with wallpaper, towels, etc.

Oh wow- very nice possibilities! Nice wall tile too. I’d paint it the same colour as your walls, some form of white. You can strip the paint off the hardware by simmering it in a (no longer used) pot with some baking soda in it.

I didn’t know the baking soda trick-nice!

So, let me summarize: Is the consensus that I go all white on the trim and the built-in, plus stripping the original hardware, and having the walls be a color besides white?

What does the rest of your bathroom look like? I’d do some nice wallpaper, and the rest just like you said. I’m no designer though. Depending on how far down the renovation road you want to go, perhaps consider replacing the floor tile to something more era appropriate? I think alot of bathrooms from this era had chrome (or something like it) hardware...but if you have original hardware that is brass, continue with that. I would think that the latches on the top cupboards will be an indication once you’ve got the paint off, whether the knobs etc should be brass or chrome(?) I’m not even sure whether you have to put the baking soda in...I’ve done it with and without and it’s worked both ways.

What is behind the little door next to the toilet 🤔

The little door next to the toilet is just storage. It’s convenient for extra toilet paper!!! :)

No, I wouldn't say there's a consensus that it should be white. You've had various opinions favoring white OR a contrasting color. It's really up to what you like. One thing I would do,though, is change that little toilet paper door to a bigger door that covers the wall. It looks really odd like that. If one door is too big, put a panel that looks like a door but doesn't open below it.