Another answer was given by Aarts and De Groot in [Complete regularity as a separation axiom][1] Canadian Journal of Mathematics 21 1969 96–105. Frink's condition sets things up for a proof a la Urysohn's Lemma; Aarts and De Groot' condition is in terms of subbases and the proof proceeds by constructing a Hausdorff compactification.


  [1]: http://cms.math.ca/10.4153/CJM-1969-010-0