Select elements in a list at level 1

I think this is what you may be after:

Select[list, #[[1, 2]] > 0 && #[[2, 2]] > 0 &]
{{{0.883787, 0.432163}, {-0.555896, 0.690923}}, 
 {{-0.416166, 0.596879}, {0.0341831, 0.000531598}}}

Select[And @@ NonNegative[#[[All, 2]]] &] @ list

{{{0.883787, 0.432163}, {-0.555896, 0.690923}}, 
 {{-0.416166, 0.596879}, {0.0341831, 0.000531598}}}

Also

Select[NonNegative[Min@#[[All, 2]]] &] @ list

Cases[{{_, _?NonNegative, ___} ..}] @ list

Pick[list, And @@@ NonNegative[list[[All, All, 2]]]]

Pick[list, NonNegative[Min /@ list[[All, All, 2]]]]

Another way:

Select[list, AllTrue[Last /* GreaterEqualThan[0]]]

{{{0.883787, 0.432163}, {-0.555896, 0.690923}}, {{-0.416166, 0.596879}, {0.0341831, 0.000531598}}}