The alternator won't have any effect on the car starting - that's the battery's job. The alternator takes over when the car is running. Now if the alternator is bad and not charging your battery, then yes, it could be the alternator. You can have the alternator tested at AutoZone for free as well, but it's gotta come out.
What happens when you go to start the car? The car will seem to lose all power and do nothing and when you turn the car off the alarm goes off? I gotta say that sounds like a bum battery - either from old age or an alternator not charging it.
Have the battery tested first and then go from there. Do the easy stuff first.