I’m not an apologist, theologian or ethicist. Nevertheless, it is my belief that most moral systems (in my mind, particularly the Christian one) have objective bases without being totally dependant on the underlying theologies.
When one asks for some “objective” basis, one is only saying he is seeking something outside his own opinion about it that plainly points to the truth of it. Well, as an example, it is not difficult to conclude that refraining from murder, and forbidding it in a society has a value independent of what I might personally think about it. Murdering is simply an unsuccessful life strategy, broadly speaking, as it very much encourages further murdering; a circumstance in which no one is safe. In recent times, we can look at the situation in Somalia or in the Tutsi/Hutu wars, and it is perfectly obvious that murdering can become endemic in a society and make the entire society unsuccessful and miserable, notwithstanding that most of the murderers subjectively thought themselves justified in doing it.
Catholics (and some others) would say that’s the dictate of “natural law”; a code imprinted on humanity as humanity, but reflecting, imperfectly, the laws of God.
But I think an atheist (and you can correct me if you disagree) would think some kinds of conduct are simply destructive of human life and happiness without significant exception, or are, at minimum, unsuccessful societal strategies, regardless of what the actor thinks about them.
If one grants that, then one has accepted that there are “objective” values that should be followed. Whether one calls them “moral” or not depends on what the source one thinks they arise from and whether that source is worthy of conforming our conduct to. An atheist might think (and I am no expert on atheist thinking) society has, after millenia of observation, determined that certain acts are properly to be encouraged or forbidden. And in that sense, perhaps, to an atheist, proscription of murder, for instance, might be considered “moral”.