It is difficult to think of a solution that does not over-enumerate, except perhaps that no subsequence that begins with a negative number can be a candidate. So here is an exhaustive approach, which could be made much less understandable by more creative coding. ;-)
In[1]:= data = {1, 2, -4, 1, 3, -2, 3, -1};
In[2]:= (* start and end of all possible subsequences *)
indices =
Flatten[Table[{i, j}, {i, Length[data]}, {j, i, Length[data]}], 1]
Out[2]= {{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1,
8}, {2, 2}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {2, 8}, {3,
3}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {3, 8}, {4, 4}, {4, 5}, {4,
6}, {4, 7}, {4, 8}, {5, 5}, {5, 6}, {5, 7}, {5, 8}, {6, 6}, {6,
7}, {6, 8}, {7, 7}, {7, 8}, {8, 8}}
In[3]:= subsequences = Take[data, #] & /@ indices
Out[3]= {{1}, {1, 2}, {1, 2, -4}, {1, 2, -4, 1}, {1, 2, -4, 1, 3}, {1,
2, -4, 1, 3, -2}, {1, 2, -4, 1, 3, -2, 3}, {1, 2, -4, 1, 3, -2,
3, -1}, {2}, {2, -4}, {2, -4, 1}, {2, -4, 1, 3}, {2, -4, 1,
3, -2}, {2, -4, 1, 3, -2, 3}, {2, -4, 1, 3, -2, 3, -1}, {-4}, {-4,
1}, {-4, 1, 3}, {-4, 1, 3, -2}, {-4, 1, 3, -2, 3}, {-4, 1, 3, -2,
3, -1}, {1}, {1, 3}, {1, 3, -2}, {1, 3, -2, 3}, {1, 3, -2,
3, -1}, {3}, {3, -2}, {3, -2, 3}, {3, -2, 3, -1}, {-2}, {-2,
3}, {-2, 3, -1}, {3}, {3, -1}, {-1}}
In[4]:= sums = Total /@ subsequences
Out[4]= {1, 3, -1, 0, 3, 1, 4, 3, 2, -2, -1, 2, 0, 3, 2, -4, -3, 0, \
-2, 1, 0, 1, 4, 2, 5, 4, 3, 1, 4, 3, -2, 1, 0, 3, 2, -1}
In[5]:= (* large to small *)
orderingOfSums = Ordering[sums] // Reverse
Out[5]= {25, 29, 26, 23, 7, 34, 30, 27, 14, 8, 5, 2, 35, 24, 15, 12, \
9, 32, 28, 22, 20, 6, 1, 33, 21, 18, 13, 4, 36, 11, 3, 31, 19, 10, \
17, 16}
In[6]:= (* subsequence sum pairs *)
subsequencesAndSums = {subsequences[[orderingOfSums]],
sums[[orderingOfSums]]} // Transpose
Out[6]= {{{1, 3, -2, 3}, 5}, {{3, -2, 3}, 4}, {{1, 3, -2, 3, -1},
4}, {{1, 3}, 4}, {{1, 2, -4, 1, 3, -2, 3}, 4}, {{3},
3}, {{3, -2, 3, -1}, 3}, {{3}, 3}, {{2, -4, 1, 3, -2, 3},
3}, {{1, 2, -4, 1, 3, -2, 3, -1}, 3}, {{1, 2, -4, 1, 3},
3}, {{1, 2}, 3}, {{3, -1}, 2}, {{1, 3, -2},
2}, {{2, -4, 1, 3, -2, 3, -1}, 2}, {{2, -4, 1, 3}, 2}, {{2},
2}, {{-2, 3}, 1}, {{3, -2}, 1}, {{1}, 1}, {{-4, 1, 3, -2, 3},
1}, {{1, 2, -4, 1, 3, -2}, 1}, {{1}, 1}, {{-2, 3, -1},
0}, {{-4, 1, 3, -2, 3, -1}, 0}, {{-4, 1, 3}, 0}, {{2, -4, 1, 3, -2},
0}, {{1, 2, -4, 1},
0}, {{-1}, -1}, {{2, -4, 1}, -1}, {{1,
2, -4}, -1}, {{-2}, -2}, {{-4, 1, 3, -2}, -2}, {{2, -4}, -2}, {{-4,
1}, -3}, {{-4}, -4}}
In[7]:= max = subsequencesAndSums[[1]]
Out[7]= {{1, 3, -2, 3}, 5}