How about
f[{a_, b_, c_}] := Quiet[Check[NMinimize[a x^2 + b x + c, x], NMinimize::cvdiv]]
l = {{1, 2, 3}, {-1, 2, 3}};
r = f /@ l
which now returns
{{2., {x -> -1.}}, NMinimize::cvdiv}
So you can simply discard all theNMinimize::cvdiv entries. You can also use your own error message
f[{a_, b_, c_}] := Quiet[Check[NMinimize[a x^2 + b x + c, x], "opps", NMinimize::cvdiv]]
returns
{{2., {x -> -1.}}, "opps"}
Now Cases[] can be used to select the good results if needed. The position of each the "opps" or NMinimize::cvdiv is the same as the data position. This way you can also know which data failed entry failed. You can use `Position` to determine this.
