Which option is better for readability when dealing with blank strings?
Instead of returning a "        " or string.Empty.PadRight(8, ' ') you should extract the values to some meaningful constants.  

The easiest way would be for your example to just use the overloaded constructor of the String class like  

private const string eightSpaces = new string(' ', 8);  


this makes it clear for Mr./Mrs.Maintainer what it stands for. But because this seems to produce CS0133: The expression being assigned to &lt;constant&gt; must be constant. you should just use a readonly string like  

private readonly string eightSpaces = new string(' ', 8);  


Using your first version will lead for each bug which needs to be solved to count the spaces which are returned.  

The name of this property is also poorly named. It doesn't represent an ImmediateDestinationRoutingNumber but instead ImmediateDestinationRoutingNumberSpaces.