We don't apply a background color to that notice. Nor does it have a CSS class around it specifically so you won't be able to style it unless you completely override the language string and just add your inline style there since you want it to apply to emails too. Example as follows with language override.
Key: UE_SENDPMSNOTICE
Text:
Code:
<div style="background-color: red;">NOTE: This is a message generated automatically by the Connections system. It has the connecting user's address, so you can conveniently reply if you wish to.</div>
Note this will have no impact on already stored private messages or sent emails.