How to cite Babai's new graph isomorphism result?
First off, I would discourage submitting for publication an unconditional paper which depends on the quasi-polynomial result, if that's what you want the citation for. Rephrase the result as conditional upon the the existence of a quasi-polynomial GI algorithm and state in a footnote that Babai may have proven this but that the paper is not publicly available. In this case, no citation is necessary because you don't need the result for the paper.

In any other context, I don't think it's particularly necessary to cite an available paper - citing his website is fine. It depends a little on what you are writing, but I would recommend asserting something along the lines of "it is widely believed that GI is solvable in quasipolynomial time and a proof of this has been announced by Laszlo Babai [citation to the webpage where he makes the claim]."

One notable advantage of citing his online claim is that his website contains both his own words as to the current claim and a link to his preprint.