To some extent, it depends on their promoters and users. If the organisers of a project like this promote it by advertising on common topic forums and websites, then only those already involved in the topic will know about it. It also depends whether the users are interested enough to promote it to their friends who may not be directly involved in the topic.
There may, however, be other opportunities for the project organisers to get some interest. At a quick look at the Galaxy Zoo website, it seems to me that it could be quite educational. A nicely phrased email to a school or two might persuade them to consider using it in their curriculum, or encourage pupils to contribute. I, for one, wouldn't be able to classify galaxies according to their shape because I don't know what galaxy shapes are called, but it would be a good opportunity to learn.
Another source of public contact is through news and the media. Occasionally, a project like these may make a new discovery, however minor, which can be reported in the news. Traffic stats for the weeks after the report will show increased visits, which means more eyeballs on your work. You can conclude from that, that these public science project do increase public interest when they do something big.
In a bit of searching, I found a paper: "Investigating public science interest and understanding: evidence for the importance of free-choice learning", John H. Falk. This quotation is from the abstract:
A random telephone survey of Los Angeles, California residents found that nearly half (43 percent) of the public's self-reported science understanding derives from leisure time, free-choice learning; science understanding was primarily acquired for reasons related to personal interest, need and/or curiosity
I quote this because it tells me that if people are interested, they'll learn. The relationship there is that if you have someone who thinks astronomy is an interesting topic, and encounters Galaxy Zoo, they'll get into it. In doing so, they'll start telling people about it, and getting others interested.
The conclusion I draw is yes - dependent in part on how the project in question is promoted.
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