Adam- Today() is a Date field whereas Now() is a datetime field (accurate to seconds level). Unless your date dimension is also a datetime field the Now() logic won't work as it supposed to. Using your dataset I've created two calcs, both returned same number of customer_Ids.
As of Now,
COUNTD(IF [Date] <=NOW() THEN [Customer Id] END)
As of Today,
COUNTD(IF [Date] <= TODAY() THEN [Customer Id] END)
Hope this helps.