One of the major criticisms against greetings is that, since the question preview isn't very long (~200 characters, I believe), greetings which inevitably take the first x characters of that preview just make it harder to recognize a good question from a bad one without needing to first load it in full. It should be noted that this argument is often defended when it regards common English greetings, which are far shorter than the full basmalah.
That argument doesn't really apply to answers though, since (as far as I know) they don't have a preview. Adding a greeting thus merely adds an extra line or two to the page. This is not only (mildly) annoying because it just makes the page longer for no good reason, but it also has the additional problem of just looking...unprofessional.
That said, there more than sufficient Islamic basis for praising God before partaking in such writings. There is also more than enough precedent in existing Islamic Q&A sites for including the basmalah and other forms of God-praising; unlike the salam — as a greeting it tends to make the site appear like a typical forum and/or discussion board which the Stack Exchange model very much isn't — I feel that adding such praises would not detract from the professional appearance of the site at all as long as they're of reasonable length and the post otherwise answers the question well.
Scholars on a number of professionally-run and successful Islamic Q&A sites (e.g. IslamQA, Qibla Answers) frequently open with a form of praise such as the basmalah or al-hamdulillah, and close along the lines of "And Allah knows best." Even then the opening and closing praises often only take an additional line each, while surrounding a substantial answer.
While I would still personally recommend against inclusion, just for the sake of brevity and making the actual answers that much easier to find, if the poster chooses to couch his answer in such forms of God-praising (as contrasted with greeting) I feel that that shouldn't be discouraged, at least not to the point that they be edited out after the fact.
However, I think they should still fall under the general "no greetings" guideline for questions due to the preview-space issues mentioned earlier.
- Questions: Do not include.
- Answers: Include at poster's discretion so long as they…
- are short
- look professional
- don't distract from the answer