We just came back from a 7 day trip last week. My DD is 21 months - we watched the Mickey Mouse Clubhouse cartoon to get familiar with the characters and then watched the DVDs that had 'real' characters on it before we left. We also prepped her by telling her she could talk to Mickey, and that he was big but not scary etc.
My DD has hated the Easter Bunny, Santa, and Chuck E Cheese so far...
We had two character meals, one the first night we were in Disney. She LOVED them. We did CP with Pooh and friends. We saw Tigger in Magic Kingdom the next day in Pooh's Playful Area and while she left DH carry her up to him, she really wanted nothing to do with him. I started to think the first night was a freak thing and that she would be afraid at our next character dinner but she wasn't. The only thing we can come up with is that when she was in her high chair, they lowered themselves to her and she didn't feel put on the spot nor did she have to actually approach them.
My DH thought some of it had to do with the fact that she 'knew' the characters and with Chuck E Cheese at least she had no idea who he was really, so no association other than some big scary thing. She knows Minnie and Pooh and stuff to be friendly, likeable things.
We figured if she was scared we could probably sort of hold her when the characters came near so she didn't get surprised. Everyone told us that the characters sense when to not come up to the kids etc, and that you can still do the dinner and just not have the characters come over. We had our two older godchildren with us (5 & 9), so we sat them both on one side of the table, and my DD on the other - this way the characters could go to them and DD could just look across the table.
Good luck!