Question: I have an object with a high non manifold edge count. Is there a way to resolve them automatically or a way to make it easier to solve them manually?

Some of them are edges without a face and others are cubes connected to the object by one edge.
The object have a high polygon count because it is the result of various modifiers applied.

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Edit:

When I delete those edges they leave others "non manifold edges"
![enter image description here](https://i.stack.imgur.com/bAldR.png)
After deletion, the solid is open and some new non manifold edges are formed
![enter image description here](https://i.stack.imgur.com/RTIka.png)
Using make faces creates some non manifold vertices
![enter image description here](https://i.stack.imgur.com/GPQHA.png)

Answer: 
Re-topology is the best option, since it will generate a clean mesh for 3d printing and will consume less time then removing unwanted cubes, edges etc.

In given situation, I'll first delete half of my model and add a mirror modifier to it. covert my mesh to Quads and remove extraneous edges and vertices manually.

Use BSurface Addon for Re-topology, as it is the fastest method.
If your concerned about non-manifold edges then you plan to 3D print this model. In that case I would add some "material" to the edges being joined. By that I mean some solid printed material that will hold the pieces together rather than a flimsy paper thin join that can break easily.

Start by separating the joined edges. Add a small bevel `Ctrl``B` so that the new edge is wide enough. You may want to display edge sizes while doing this, see Mesh display in the properties panel. The width you use here will depend on the material you plan to use, the width of the new edge added by the bevel should be at least the minimum material thickness you can use. Repeat for the other edge.

![enter image description here](https://i.stack.imgur.com/mCLd1.jpg)

Select the vertices or edges around both new beveled faces and bridge edge loops (in the specials menu `W`)

![enter image description here](https://i.stack.imgur.com/XKK16.jpg)

This will remove the non-manifold edges and leave you with a solid join in your printed model.

![enter image description here](https://i.stack.imgur.com/uw6dG.jpg)

You may want to bevel these new edges to get a more rounded join.

Another option is to move the cubes together so that both beveled edges meet up and then remove doubles. This will effectively make the corner of the cubes intersect a little and depends mostly on your needs. If your using the array modifier then you can intersect each cube by using and offset a little less than 1.0, I'm not certain if this overlapping non-connected geometry is acceptable for 3D printing so you may have to manually join the overlapping areas.