Qualifications
To be a curator, you have to be smart. That means a lot of school. Most institutions will want you to have at least a master's degree in a related field—some may even want you to have a doctoral degree (usually for such places as natural history or science museums).
You may be able to get away with having only a bachelors if applying at a small museum, for example, but then you can expect to also be sharing many of the business administration and/or public relations duties, so you should be well-versed and educated in those areas, too (source).
You should have a genuine love of your area of study and of studying in general, because a lack of passion will show through your work. So if you're one of those nerds (we use the term endearingly) who does extra credit even when s/he's already on pace for an "A" and writes essays the night they're assigned, this might be all you.