Any way to fix that?

On the monitor the colour is a nice black - when printing on an Epson Workforce 610 that background is a dark royal blue.
Colour settings in the print menu don't seem to show in the print preview picture, so it's not really clear what to adjust and by how much.

The print preview picture btw looks pretty much black.

The skin tone of the printed picture with the dark blue background goes a bit into the pink compared to the original so I was going to reduce the Magenta in the printer menu to see if that has an effect. I just don't want to waste paper and ink experimenting too much.

Any suggestions how to best attack this problem?

Start by doing the color sync routine for your monitor.

Hopefully still found System Preferences>Display>Colour>Calibrate. Setting the desktop to a neutral gray colour will make it a bit easier to determine if the sync is out of whack. Be sure to set the gamma to PC standard as this is what the printer is probably expecting to see.

For printing, calibration and a good monitor are key. It could be that the monitor just isn't calibrated and it could also be that the monitor just isn't capable of showing the detail in the shadows that exist.

You can use the built-in tool in OSX, but you will get better results with something like the Spyder or other calibration tools. They are expensive, though.