You need to set the UV map of the object. This establishes the relationship between your map or texture (which are 2D) and your object (which is 3D - even if it is flat!). In your illustration, a number of points are all at the same point on the UV map, hence the same color is applied to them, and that is what is creating the stripes.
There is a UV mapping tool in the vertex modeller (three icons at top right, it is the third one). For a simple shape like the one in your illustration, a planar mapping face on to the object will work fine and give you the results you need. UV mapping for complex organic shapes requires a much longer explanation!