How to remove a line of a matrix if not all elements are numeric?

The following should work:

lis = {{1, 2, 3, 4, 5, 6}, {1, 9, , 4, 6, 2}, {4, a, 3, 7, 1, 2}, 
      {3.4, 5.2, 6.5, 7.7, 6.1, 2}};

Then:

Cases[lis, {__?NumericQ}]

{{1, 2, 3, 4, 5, 6}, {3.4, 5.2, 6.5, 7.7, 6.1, 2}}

We can also use VectorQ with Select

Select[lis, VectorQ[#, NumericQ] &]

If you have Version 10, the following works:

Select[lis, AllTrue @ NumericQ]

m = {{6.22926, 6.23254, 6.23128, 0, 6.23453, 6.22548, 6.22938, 
   6.23443, 0}, {6.22926, 6.23254, 6.23128, , 6.23453, 6.22548, 
   6.22938, 6.23443,}}

Select[m, MatrixQ[{#}, NumericQ] &]

{{6.22926, 6.23254, 6.23128, 0, 6.23453, 6.22548, 6.22938, 6.23443, 0}}

more alternatives for speed comparisons.

m = {{1, 2, 3}, {, 2, 3}, {1.1, 2, 3}, {1.1, , 3}}

DeleteCases[m, 
 Alternatives @@ Select[m, MemberQ[#, _?(Not[NumericQ[#]] &)] &]]

{{1, 2, 3}, {1.1, 2, 3}}

Edit

A timing comparison. First creating a matrix:-

m = RandomReal[1, {10000, 10000}];

(* Adding some blanks *)
(m[[Sequence @@ #]] = "") & /@ RandomInteger[{1, 10000}, {2000, 2}];

Comparing the various solutions:-

First@Timing[a = Cases[m, {__?NumericQ}]]

19.141323

First@Timing[b = Select[m, MatrixQ[{#}, NumericQ] &]]

0.468003

First@Timing[
  c = DeleteCases[m, 
    Alternatives @@ Select[m, MemberQ[#, _?(Not[NumericQ[#]] &)] &]]]

85.862950

a == b == c

True