Should computers be shut off during a thunderstorm?

Just shutting down your computers offers little protection from a lightning-induced power surge. As in most modern electronic devices, power is supplied to the circuitry of a PC as long as it is plugged in, so an electrical surge of sufficient intensity can still run through the machine even when it appears to be off. Some PCs do have a mechanical on/off switch on the power supply (in the back, near where the power cord plugs in) and this would in fact break the circuit. Even so, if lightning were to actually strike your power lines (as opposed to inducing a surge) the current would easily jump that small mechanical gap and electrify the PC.

Lightning can also strike and travel through your phone wiring, which is why you are warned to stay off the phone during a storm. The danger is somewhat mitigated in a non-residential setting, where the switching equipment may be equipped with a suppression device, or otherwise act as a very expensive circuit breaker between the outside line and the phones. Even so, it's a good idea to stay off the phone during a storm. (It is safe to use a cordless phone. A spike may travel into the base unit, but it can't "jump" to the phone without a physical connection.)

Do you need more to worry about? Even a distant lightning strike can produce an electromagnetic pulse sufficient to induce a significant current in your phone or network wiring. While not a safety concern, such a surge can "fry" components such as network cards, routers, switches, and modems (i.e., fax machines).

So, to absolutely protect your equipment, you would not only have to shut down all your electronic devices, but unplug every power, network and phone cable going into them. This is clearly impractical, if

The best protection against these threats is to install surge suppression on all potentially affected devices. An all-purpose suppressor will accommodate your phone and network connections as well as the power circuit. Realize too that not all suppressors are created equal. Among other things, better quality (i.e., more expensive) units react to spikes and surges more quickly in situations where just a few milliseconds can make a difference. Also, these suppressors have higher-end components that tend not to degrade over time as do the "supermarket" variety power strips.

With high-quality surge suppression as a standard feature, uninterruptable power supplies (UPS) provide additional protection for some equipment. A power outage can be as damaging as a surge, especially to computer hard drives. If power is lost while the hard drive is writing data to disk, not only is unsaved data lost, but open system files can be damaged, and in some cases the entire file system can be permanently corrupted, rendering the system unbootable. This sort of damage can occur merely with a sufficient drop in voltage as well as a total power loss. When any such voltage fluctuation occurs, a UPS will instantly switch to battery power to keep the equipment running.
Providing a UPS for every device in your library may not be economically feasible. For printers, fax machines, copiers, and even public access computers, surge suppressors should be sufficient protection. UPSs are advisable on staff PCs and network switches (hubs). During an outage, the UPSs will keep the PCs running and connected to the network long enough to save work, close programs, and perform "proper" shutdowns. Many UPSs now have "smart" USB connections to the computer; using either the manufacturer's utility or Windows' built-in UPS service, the UPS can "talk" to the computer, and initiate an automatic shut down after some pre-specified time, or at some percentage of remaining battery life.