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I would make this mesh's extrusion be at about a 45° angle instead of a 90° angle.
Taking the bottom vertices and scaling them kind of gets what I want, but many vertices go the opposite direction that I want them to go, because every vertex is moving in relation to the center of the object:
How can I make the bottom vertices go like this?
To scale something in the direction of the normals, you can use Alt + S
Using this tool, you can fatten a model or make it more slender.
For a model that has many thin parts that are close together like this one, there will likely be some overlapping parts when you fatten it so a bit of manual work will be needed to clean it up.
Using a LICEcap screen-capture .gif I show an easy way to clean up the overlapping edges. I'm using the circle select tool C to select all the edges then with the selection tool still active use MMB to deselect the surrounding edges leaving only the one which is not visible. That one edge is not visible because it's facing directly in and out of the screen and the display cannot handle that situation but the line is still selected even though we can't see it.
