Apply conditional formatting with custom formula
=arrayformula(sum(N(not(isnumber(A3:A)))))=0
This can be applied even to a range of headers such as A1:E1; the formula, entered as above, will be interpreted in a relative way for other columns.
Explanation
not(isnumber(...))
is true if the entry is not a number
N
converts logical true/false to integers 1 or 0
sum
adds these.
arrayformula
makes sure the entire range is handled
- the sum is equal to 0 if all entries are numbers.
Note
Dates and times are considered numbers, because internally, they are. For example, the date 05/26/2016 is the number 42516 formatted as a date.