Stax, 9 bytes

ΘK♂▓b♪Q4⌠

Run and debug it

This is a packed Stax program, representing the following 11 byte version:

Stax, 11 bytes

c{|+}mc|mI@

Run and debug it

Explanation

c{|+}mc|mI@ # takes an array of strings as input
 {|+}m      # map sum across all strings in the array
       |m   # get the minimum
      c  I  # get the index of the minimum in the array of sums
c         @ # get the value at that index in the original array of strings

Stax, ⁹ 5 bytes

And pure ASCII!

{|+}e

Run and debug it (with test cases)

Explanation

{  }e # minimum by
 |+   # sum