Does an enlarged goiter affect your voice?

Thyroid effects. There are a few ways the thyroid can affect the voice. If it's underactive, you can get thickening of the vocal folds which will change the voice. Also, thyroid cancer can invade into the nerve that controls the voice so it can do it, too. Finally, a vrey big goiter can compress the nerve leading to bad function. A large thyroid can also affect swallowing.
It can. The voice box (larynx) is a very sensitive organ and it is easily affected by many problems (allergy, acid reflux, overuse, thyroid enlargement, throat cancer, etc). Voice problems that are severe and/or longstanding should be evaluated by your local ENT or laryngologist.