For overcurrent faults, it was simply considered an acceptable risk in most cases...
For many 240V and 120/240V devices, such as large appliances and heaters, the risk of the device being still "live" if an overcurrent fault was present was considered acceptable then, as the fault would not be fed power from the other leg. The only case where it could be is if the circuit had multiple loads of different voltages (L-L, L-N, L-L-N, 3P/3P+N) on it, and that situation is generally rare, even today.
Also, as Ed mentions, controls for single-phase 240V heaters and such often only switched one hot wire. This was, and is, acceptable because the control means isn't considered a disconnect in this case, and may not even have an "off" position or function to begin with.
...but in the cases where it wasn't, it was dealt with differently
However, the above logic fails to hold for one very important class of devices, albeit a class not found often in houses: polyphase motors. In a three-phase motor circuit, if a single phase fuse blows due to a fault, the motor will continue to run on the remaining phases, drawing more current from them while back-energizing the fault in the fashion of a rotary phase converter. Worse yet, the types of fuses commonly used then were not as refined as today's models, and thus could not be as easily sized to provide overload protection for a stalled motor.
Enter the electrical circuit protection device that no residential electrician ever sees: the motor overload relay. These devices are sensitive overload (but not short-circuit) protectors that break the control circuit when opened, and are connected in series with the motor's control contactor and a manual disconnecting means in industrial applications that use separately protected motors, forming a motor starter stack. The motor control centers found in plants and other industrial-ish buildings consist of "buckets" that contain a motor starter and local control devices for that motor; a more sophisticated system might have a cable running off to a PLC or control panel for the machinery that motor is part of.
Because these devices are set for the specific motor they are used with, they can be set up to be much more protective than fuses were back then. Poly-phase overloads also provide a common-trip type of functionality that opens the control contact when an overload is sensed in any phase, thus disconnecting the motor entirely and preventing "single phasing" as could happen with fuse-only protection schemes. In addition, the controls on such a motor would be set up for an "undervoltage release" that prevents the motor from restarting automatically if the power "blips" out, preventing injurious surprises from unexpected machine restarts and further mitigating against single-phasing of the motor from a blown fuse.
Note that domestic "bare motor" applications, and just about all appliances, use motors that have the thermal protector built into the motor, which is why you don't see such elaborate motor protection and control setups in homes and appliances. A few very old setups may have used manual motor starters, which are essentially a combination of a lightswitch and an overload coil that can turn the lightswitch off if the connected motor is overloaded. These are rarely used, though, as they do not provide any facility for automatic or remote control of the motor in question.
Disconnecting was handled via pullouts or safety switches
Of course, none of this helps you when you're trying to turn the darn thing OFF to work on it, or change a blown fuse for that matter. This requires an all-pole switching means to be present at the fuses, and this took the form of either a pullout disconnect or a fusible safety switch. Both are still made and used today, even, despite the ubiquity of circuit breakers nowadays.
Pullouts are easy
Pullout disconnects are simple devices at heart, consisting of a line and a load set of switch jaws, mounted to an insulating base and connected by a pullout plug that contains mating sets of blades connected either by busbars or by fuseholders + cartridge fuses. The base has sidewalls that guide the plug in and out as it's installed and removed, resulting in a simple, inexpensive, yet surprisingly reliable disconnect for low-usage applications. This is part of why they have persisted to this day for air conditioner maintenance disconnects, but their low cost and ease of procurement also are good for applications such as branch-circuit disconnecting for outbuildings.
If you're wondering where all those old knife switches went, though...
The other type of fused switch commonly used in light-to-medium duty applications (I'm omitting heavy-duty creatures like bolted pressure switches from this discussion) is what is called an enclosed safety switch, or simply "safety switch" for short. They are the direct descendants of the old, open-blade mains knife-switches of Victorian days, simply with the blades wrapped in a box that has an operating handle on the outside, hence the name "safety switch" since you can't poke their conducting bits with your finger unless you make a concerted effort to do so. You can still buy them, as well; while they are more expensive than a pullout switch, they are available with much higher current ratings, and in a variety of enclosure types, allowing their use in a wide variety of commercial and industrial applications.