I'm assuming you have a no-crank-no-start, and not just a no-start condition?
Confirm that the battery is at 12.4-12.6 volts or above. Confirm that you have battery positive on the alternator's terminal. Confirm that the alternator is properly tightened down.
IF your battery is still good and charged, a bad alternator will not cause a no-crank-no-start condition. The only reason an alternator would cause a no-crank-no-start or an extended crank is if you drove the car with the bad alternator, draining the battery.
Do you hear any clicks when you try to start it? Maybe one loud click? A series of quiet clicks? Also, it would be a good idea to clean ALL of your ground connections, your battery posts + terminals, and to test the starter.
EDIT: Measure the voltage at the battery. It should dip no lower than 11 volts when you try to start the vehicle (turning the key past accessory position.)