Efficiently select the smallest magnitude element from each column of a matrix

m = 
  {{-351, -260, -148, -159, 1}, 
   {-197, -106, 6, -5, 155}, 
   {-194, -103, 9, -2, 158}, 
   {-104, -13, 99, 88, 248}, 
   {28, 119, 231, 220, 380}};

Lets not forget that Abs is Listable, so we only have to map Min.

Min /@ Abs[Transpose[m]]

{28, 13, 6, 2, 1}

This

mat = {{-351, -260, -148, -159,   1},
       {-197, -106,    6,   -5, 155},
       {-194, -103,    9,   -2, 158},
       {-104,  -13,   99,   88, 248},
       {  28,  119,  231,  220, 380}};
Map[Min[Abs[#]] &, Transpose[mat]]

finds the minimum absolute value element in each column {28, 13, 6, 2, 1}

You can do the same thing without needing to understand # and & by

findminabs[v_] := Min[Abs[v]];
Map[findminabs, Transpose[mat]]

You can also use Composition (@*)

Min @* Abs /@ Transpose[mat]

{28, 13, 6, 2, 1}

To retain the signs:

MinimalBy[Abs] /@ Transpose[mat] // Flatten

{28, -13, 6, -2, 1}