Use conditional formatting.
If the 'other cell' with a date in it is cell H4, then:
Select cell G4
From the Ribbon select Home - Styles - Conditional Formatting,
From the drop down select 'Manage Rules' and select 'New Rule'
In the dialog box select the last item in the list 'Use a formula to determine which cells to format'
Enter this formula in the box
Note that TRUE and FALSE do not have double quotes - they are Excel's logical values true and false.
Click the format button and from the 'Fill' Tab select a red color, click OK
Click OK again to exit CF.
Cell G4 will now change color based on the date in cell H4.
Note that the Integer function in the formula INT(H4) removes any time information from cell H$, so that only the date information is tested against Today's date.
Excel stores time in the decimal part of a number and date information in the integer or whole number part.