You can click on the Label shelf of you mark and select Alignment - Vertical and change Automatic for Center .
As an alternative, assuming that your attribute is a dimension then instead of using the measure alone you could do something like:
IF TOTAL(COUNTD([the attribute])) > 1 THEN
[your aggregate measure]
And put this on the Label Shelf. The Compute Using would be along the attribute dimension. If you're using Excel or a data source that doesn't support COUNTD(), then you could use SIZE() instead of the TOTAL(COUNTD()).
Technically the ELSE part isn't necessary, it's just to highlight that if you set a specific measure result to Null then Tableau won't display a label.
I can't believe it was this simple - I tried changing alignment in various spots but must have never selected the vertical alignment. Thank you very much!