How to tone down the color?

Yes you can use a product like Rustoleum Transformations antiquing gel. You paint it on then wipe it off with a rag. It will give you a darker color.

This is a technique called black washing, it may work for your hutch, and you can control how much or little you may want to tone it down:

https://salvagedinspirations.com/how-to-create-a-black-wash-paint-finish-painted-nightstands/

Fusion paints has a Stain and Finishing Oil that you can wipe on over finishes to change the color

Great ideas about antiquing it. My first thought was to put wallpaper or some type of backing on the back wall.

Use an antiquing medium. Wipe or brush it on and then wipe off as much as you like.

Wallpaper on the back wall would be very pretty. I think to tone it down I would use a white wash. We would love to see what you decide to do with it!

It could depend on your aesthetic do you want to darken it muddy up the color with a brown or a black like an antiquing wax or white wash it with the white thinned paint. There Is to be crackle finish as well. Some more information on the exact look you prefer could be helpful. Using the perfect product for your aesthetic is recommended.

chalk paint over it with a light grey or white

Without completely repainting it. You can mix 50/50 water and paint in a darker shade and white wash it. Use a cloth to remove and blend the paint.

I love the idea of just adding wall paper to the back of the unit. This would pull you eye away from the blue and focus it on the wall paper or contact paper depending what you use.

If you use the right wall paper it will soften the look of the blue on the wood.

If you paint the shelves that would also soften the blue. The color you paint the shelves could coordinate with a color from the wall paper on the back of the unit.

I agree with Mogie that wallpapering the back walls (and adding shelf liner to the shelves) would do wonders to tone down the bold color and soften the piece overall. If you are not experienced with refinishing furniture, it might be an easier and faster option to repainting the entire cabinet.

Here are a couple of examples for inspiration.

Antiquing wax will make it a good bit darker. Try a bit under a shelf or back edge to see if you like it. Purchase it any place that sells chalk paint.

White wash it

You could prime over it and start over with a new color or exact color you want.

You can add a dark glaze or wax to it. Also, putting contact paper or removable wallpaper on the back of each shelf would completely change the appearance.

Yes Antique Wax!

I would lightly sand it and use a dark wax like Beeswax on it.

Sand it and repaint it white

Add some highlights with paint or sand it down a bit