This issue is solely iOS 7. In iOS 6 all you had to do was clear the Safari cookies and history. Apple must have changed something in how they store if an app banner has been closed in iOS 6 vs iOS 7. Surprise!
The following steps are what I have determined gets the smart-app banner to appear again, confirmed on a device running iOS 7.0.4:
- Settings -> Safari -> Clear history
- Settings -> Safari -> Clear cookies and data
- Settings -> General -> Reset -> Reset Location & Privacy
- Settings -> General -> Reset -> Reset All Settings
Not the cleanest, but it gets the job done.