The entire point of an internship is to get experience, and it is typically understood that it will be unpaid. While paid internships should certainly be the norm, going without pay should not be the single deciding factor when it comes to deciding whether or not to pursue an internship that will further one's career.
You will get put through the ringer as an intern and have to do a lot of tasks that make you say "why am I here" but they are a great foot in the door opportunity. When you don't have actual paid experience in an industry, the next best thing to have is exposure and people seeing you in the environment. If you do well it should end up paying off.
Although there are some unscrupulous organizations out there that either work interns to death or never give them good learning experiences, most unpaid internships function as a foot in the door, a chance to gain skills, and a way of getting exposure to explore a new environment. Most of the time it is a fair barter.
Yes, unpaid internships are worth the experience. Any practical experience that someone gets is worthwhile and certainly helpful to a future job search. But I know of a lot of people who have taken unpaid internships and have worked very hard, and that kind of overuse of people seems more like exploitation.
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