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Assume that a student has a continuous involvement in any community of Stack Exchange.(consider 1,000+ reputation).
Does It has a positive impact to mention this involvement in order to show you are passionate about that field? I am asking this because the Statement of purpose has a great impact on your MS/Phd application.
My rough answer would be that you need to convey to your reader:
Personally I'd view it as a plus but not a massive benefit. It would be more remarkable at undergrad entry to show engagement in the subject, but at masters/PhD level I'd assume you're already engaged. It does show interest in teaching which could be positive (eg if you're likely to be a TA at some point).
I'd weigh carefully what you convey by this statement against the space needed to convey it: it may be there's a more impactful way to use the space.
[Disclaimer: I come from a UK perspective. Other countries may be different]
I would argue that in itself, participation in Stack Exchange is not a major benefit nor hindrance to getting into graduate school. What is important is demonstrating that you are engaged and well-versed in the norms and culture of academia, which participation in SE.academia could possibly demonstrate. Is there some way for you to show that your participation in SE enhanced your ability to perform research, or communicate with like minded individuals? Listing "1000" Reputation on StackExchange Academia is unlikely to produce your desired results, however, a short paragraph in your Statement of Purpose, something along the lines of "Engaged with diverse, multi-national academic populace through popular website 'Stack Exchange Academia' for over 2 years helped me to develop the professional communication skills necessary to become a highly effective researcher, as well as developing a professional reputation as someone with high-quality questions and problem solving abilities".
From a US perspective too, listing a 1000 rep is unlikely to be helpful. Something that goes onto your SOP should probably involve hundreds of hours of dedicated work; it should be hard and compelling. Perhaps you could weave it into your technical narrative-- listing a discussion you had here that is particularly relevant to the rest of your SOP.
Academia.SE would probably not help at all, unless your PhD was centered around education. I would suggest that you weave it in only if you can find some posts which were significant contributions on the Electrical, Computer Science, Math, Physics SE (basically the area of your PhD).
Most programs do allow you to specify a web page (though these are often ignored). If you're planning to include one, you could add a SE flair and link it to your SE network profile.
EDIT: Yay!
