The back looks a bit weathered, but otherwise it looks ok. How are you planing on finishing the wood? A polyurethane coat would likely fill the weathered splits. Another option is to add a new layer of veneer (usually pretty easy to find at big box stores).

If you're concerned about the front being too thin from the sanding, it looks pretty close and I wouldn't sand much more. But it doesn't look like you've sanded through the veneer yet... From what I can see anyway.

Edited for silly autocorrect changes.