The corp does not have any bad publicity tokens and plays the "The All-Seeing I". Can the runner prevent resources from being trashed?
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.
All-Seeing I just gives an extra option for how to stop the trashing, it doesn't in any way stop the runner from using it's own prevention effects. If your question is can the runner chose to use the remove 1 bad pub when there is no bad pub to remove, the answer there however is definitely not.