Can the Runner prevent resources trashing via the "The All-Seeing I" if there is no bad publicity?
Yes. Spending the bad publicity is a way to prevent the total effect of All-Seeing I, but it does not state that the trashing cannot be prevented by other means. Cards like Fall Guy and Dummy Box are still able to prevent some of the resources from being trashed, even though they are themselves resources; this ruling can be applied to this situation. 

It's also worth noting that if you have a Jarogniew Mercs in play among other resources, it would be left standing after All-Seeing I resolves.