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Great question. And I appreciate how you are trying to figure out what the heck happened to spur such an odd fear, but sometimes kids come up with reasons that are not grounded in reality, and therefore difficult to figure out as an adult. But be that as it may, here is how we treat simple phobias (i.e., phobias to a specific stressor, like a balloon) in children:
We do what is called counter-conditioning. The guiding theory is that one cannot be calm AND freaked out at the same time, so we GRADUALLY expose the child to the stressful item/event while cooing and soothing them simultaneously. In other words, we slowly add stress, but emphasize the calm.
So in this example, cradle your daughter so that there is a lot of body contact (but that she does not feel "trapped") with your mouth near to her ear. Then coo to her... say comforting things, share a private secret... whatever you do/say to relax her. Then introduce a balloon into the house. It should not be close, but she may be aware that it is in the same house with her. Take however long it takes to make her calm. Then move the balloon into her field of vision - but still far off (maybe in another room, but visible). Go through the relaxation exercise again. Over time (and this may take a few sessions if her stress exceeds your ability to quell it), move the balloon closer until physical contact is made.
The other piece to this exercise is that YOU (mom) must also be very calm internally. Your daughter scans you for signs of stress when she feels herself escalate, and you must be a picture of calm in order to model the lack of threat to her. Otherwise you are going to amp her up.
Best of luck to you and your daughter. If you are satisfied with the response, please hit "Accept." That is the only way I can receive credit for my answer. Thanks-