Below you can see how the original Steam for Linux appindicator icon looks like:
And this is how it will look after the changes below with the Ubuntu-mono-dark icon theme:
And with the Ubuntu-mono-light icon theme:
For Ubuntu Mono Dark (dark panels, light icons):
cd && wget https://www.dropbox.com/s/vzve0qrugp2mf5c/steam48.png
sudo mv ./steam48.png /usr/share/icons/ubuntu-mono-dark/apps/48/steam_tray.png
sudo gtk-update-icon-cache --ignore-theme-index -f /usr/share/icons/ubuntu-mono-dark
For Ubuntu Mono Light (light panels, dark icons):
cd && wget https://www.dropbox.com/s/zygq15nfa11epdu/steam48.png
sudo mv ./steam48.png /usr/share/icons/ubuntu-mono-light/apps/48/steam_tray.png
sudo gtk-update-icon-cache --ignore-theme-index -f /usr/share/icons/ubuntu-mono-light
Update: +Martin Kozub has updated Steam Skin Manager to automatically install these monochrome icons when using Ambiance / Radiance skins. See: Download Steam For Linux Skin Manager (Includes Ambiance And Radiance Skins).