Yes you can jump start a car with a bad alternator.
However if the alternator is not charging then the battery will eventually die again and the car will also shut off once the voltage in the battery is depleted enough.
So if your alternator is bad you need to replace the alternator to continue driving the car.
The alternator not only recharges the battery but also keeps the engine running by providing enough electricity to the fuel injectors, ignition coils, fuel pump etc.
When the alternator is not charging properly or at all then the engine may run rough, stall or may not run very long.
So if your alternator is bad then you need to replace it.