Perhaps something like this?
walk=
Module[{randoms},
randoms = Table[RandomInteger[{-1, 1}] , {i, 20}];
Accumulate[randoms]
];
The set or random numbers (note that there is no factor of
i as there was in your original code) gives the instructions on what to do at each step. The Accumulate function then produces a running sum of how far the walker has gone at that step.
Visualize the walk (in the y-axis direction) using something like
ListPlot[walk]
