I have no intention of putting any flavor of SE on my resume. However, that should have little weight here because I (1) have no intention of looking for another job outside my current organization before I retire, and (2) have more job-related things then I need to more than adequately fill space. So personally I'm not going to put Chemistry SE on my resume (heck, I may never update my resume period).
I regularly hire technical people, looking for technologists and scientists, so I do look at a lot of resumes. I can't claim to ever have seen an SE site mentioned on any of them. I'm really not sure how I would react to seeing it on somebody else's resume (Chemistry SE, Physics SE, whatever). I come here for a bit of fun, a break from real work, and a chance to dig a bit into something to explain it to somebody else. It is nice when somebody votes an answer of mine up, but I've had most of my enjoyment in putting the answer together in the first place. But I know that what I do here has little to do with what I do at work, from a depth of understanding point of view. My contributions here have no bearing on whether I can do my job or not.
Furthermore, you don't have to be an expert to contribute excellent answers. One of the recent issues with the moderator agreement (see Diamonds are not forever for some others) is it has led to the removal of diamond moderator status from under-age users. These users were long time, respected members of an SE site. I also recall at least one case of an excellent contributor on Physics who was a teenager in high school. So being on, contributing to, and having a high reputation on Chemistry SE is not necessarily an indicator of the deep technical expertise that I am looking for.
But, lets suppose I saw Chemistry SE on a resume. Would they include their user name, if it is not obvious? If not obvious who the user was from their name, what could I do to verify their claim? Should I even try verify their claim? What if someone claims to be @orthocresol - do I just have to believe them? Perhaps they are an excellent member in good standing, but then we are back to the paragraph above - what does that mean to their ability to perform the job I've posted. And if they are not in such good standing (poor track record on questions/answers, a bad attitude) but have an otherwise good resume, what do I do then?
In the end, for me and my hiring, putting Chemistry SE on your resume is likely a waste of space. If you are a student or very early in your career and have nothing else to use a line on, perhaps add it. But that space is better spent on giving me a good reason applicable to the job posting (the required and desired attributes) to not move on to the next resume.