Check to see if your college has any kind of internships with museums, or contact your local museum or art gallery to see if they have any volunteers opportunities. For example, if you lived in the DC area, the Smithsonian has opportunities for internships and fellowships (they pay, but not much) for all levels of students from college to graduate students. But I would start locally, and be flexible about the type of museum; you might have your heart set on working in an art museum, but gaining experience in, say a science museum would still be museum experience. Same thing with a gallery, having retail experience in a gift shop would be transferable experience.
Of course, not everyone has the luxury of being able to spend a year or even a summer volunteering, most of us have to work at least part time to make ends meet. But even if you could only give a few hours of volunteering at a local museum or historical society, that would be a help.
I would talk to your current art history professors, and see what they suggest, or talk to your college's career office. If you have a local museum or historical society, that might be a perfect opportunity for you to conduct an "informational" interview with a curator. You could also check out the American Alliance for Museum's website www.aam-us.org for more information about museum careers.
I absolutely agree with the advice from Ms. Rosenthal. I had the same experience in Chicago, trying to break into the arts/museum scene. Volunteering and interning are definitely the way to go, though most museums unfortunately don't have the funding to pay for these programs. This allows you to add projects to your resume and build your network, which will eventually lead to a paid position. You may have to hustle through some part-time work, internships and school for awhile, but it pays off!
Many of the larger institutions have competitive programs, so try the smaller ones! The city has many great, lesser-known museums. I know a lot of universities have art galleries and museums, does yours? Is there a small local gallery that you can approach about a more informal internship? It may even be answering phones and scheduling, but you'll be around for the curatorial process, mounting the exhibition, etc., and gain contacts in the field.
Also, talk to your professors. They may have connections to individuals or institutions that would put you higher on the list of applicants to these programs. Look for art history societies - these organizations frequently offer discounted student membership and are a great place to network and gain continued education through symposia (check out the Midwest Art History Society!). Lastly, when in doubt, take to the internet. Start a blog or Instagram account focusing on your specialty, or get in contact with an existing one about potentially creating some content. This has become a great way for professionals who are new to the field to get their voices out there.
You can keep learning new skills after you graduate. You never know when your current skills would become obsolete. Networking can help you as well. I have a degree in Art History with a minor in Dramatic Arts so I should know.