From the question i assume you are going to make the boards yourself, otherwise the holes would be through plated and you would only have to solder on the bottom.
You could place a restrict rectangle or polygon on the top of the board where you do not want any tracks. So this would be on top of the pins of the connector. This way the autorouter knows it cannot put a trace there on the top and is forced to use the bottom only.
See this picture, where i placed a top-restict (layer 41) rectangle on the top connector, which forces the auto-router to only place vias where i can reach them.
Hope this helps,
Guido