My niece was petrified of anyone in a costume from the time she was an infant. Clowns, Santa, Easter Bunny, the Characters at Six Flags. It was pretty intense, she wasn't just skittish, she was totally and completly terrified.
She loves amusement parks as long as the characters aren't round, but when a character is near she would cling to her uncle and be shaking like a leaf.
We had a full family trip planned to WDW followed by the Cruise when she was 8. She had still not gotten over her fear. We knew it was going to be a problem, but we talked to her and assured her that the whole family would 'protect' her. She was the oldest of five girls who would be on the trip (2 8 year olds, 2 5 year olds and 1 3 year old).
My other sister booked a Character Breakfast for our first morning at WDW (probably not a bright idea, but she doesn't always think things through), so we are lined up outside Animal Kingdom waiting to go in when she realized that there would be characters in the park and she totally freaked, major tantrum. Her mother just grabbed her and told her that she needed to trust us, had we ever let anything bad happen to her? That she did not have to go near the characters but she did have to eat breakfast with the family. She entered the park sobbing.
We got to the breakfast and talked to the CMs and explained the situation. Our group was large enough that we were at two tables. So we had one table as the character table and the other table as the no characters table. The CMs knew to tip off the characters to not come to the one table. My niece knew that the one table would be 'safe'.
The characters came out and she hid under the table. She peered out and watched her sister and cousins going up to all the characters with the other kids. She just watched. Eventually she climbed up on my lap but I could tell she was still scared. At one point she turned to me and said that she might go stand near Goofy so that grandma could take her picture. This surprised me, but she changed her mind and didn't do it, but I could see the wheels spinning. She was starting to realize that maybe the characters weren't so bad.
We got done with breakfast and we were walking through the park when the girls saw Pooh. The four nieces went running for Pooh and the scared one went running for her mother. She looked over and saw how much fun everyone else was having and the suddenly she let go of her mother, ran over and threw her arms around Pooh, almost tackling him. We were all shocked. She came back and was so proud of what she had done and was so exited. We promptly bought all the girls autograph books and for the week at WDW and the week on the cruise, my niece was more interested in the characters than in anything else. She filled two autograph books.
She just needed to work through her fear on her own. We never pushed, we promised to protect her and kept assuring her that she didn't have to be around the characters if she didn't want to be. She saw how much fun everyone else was having and how safe it seemed to be and decided on her own that things were going to be OK.