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Transpose and then Map
{{2, 1, 1}, {0, 1, 0}, {0, 0, 1}} // Transpose // Map[If[MemberQ[#, 1] || MemberQ[#, -1], x, y] &]
Or
{{2, 1, 1}, {0, 1, 0}, {0, 0, 1}} // Transpose // Map[If[ContainsAny[#, {1, -1}], x, y] &]
{y, x, x}
{{2, 1, 1}, {0, 1, 0}, {0, 0, 1}} // Transpose // Map[If[MemberQ[#, 1], x, y] &]
