The warp core isn't the only source of power on the whole ship; it's just the biggest and electroplasmiest. Starships also have fusion power plants. Y'know, those old-fashioned atom-smashing machines? They're crude, like the power-plant equivalent of two cavemen swinging wooden clubs at each other, but they power the impulse drives, and nobody's going to complain about at least being able to go somewhere when the anomaly of the week turns the warp core into a flower pot or something.

For some reason, nothing bad ever seems to happen to the fusion reactors. I guess it's because the reaction fizzling out and shutting down quietly isn't very dramatic. Fusion reactors aren't all explodey like antimatter is.

How can they still run the EMH if they don't have power for running a few fans?