What do Atheists base their morals on?
The very concept of Atheism is underpinned by an approach to life based on logical and reasonable principles. As such, an Atheists' morals are derived from these very principles. The idea of individual property rights is a logical concept which protects the individual from bodily or material harm. As such it is not logical for a society to allow acts such as murder and theft.
In Athens in 594 BC, a Greek merchant and poet by the name of Solon developed a legal structure based on logical principles which our legal systems today still emulate. This includes concepts such as property law, social democracy, fair trials and equal rights for all people. These principles were based on logical and reasonable ideas and did not require any religious or philosophical influence.
It is worth noting that it has been those laws that were not based on logical or reasonable principles but on religious ideas that have caused the most harm to society. Ideas such as Apartheid, slavery and sexual discrimination are not logical yet they have all been supported by religions and religious institutions, even as late as the 20th century. You could go one step further and compare Athens's legal system under Solon to the Israelite legal system under Moses. While they were celebrating legal concepts such as democracy, justice and equality in Athens in the 6th century BC, the Israelites across the Mediterranean were encouraging legal principles such as stoning disobedient children to death (Deut 21:18-21), excommunicating women after they had given birth for being "unclean" (Lev 12:2-5) and formalising slavery laws (Exod 21:2).
The laws in Athens were based on reason and logic, whilst the laws in Israel were based on religious instruction. It's the tale of two cities and I know which city I would have like to have lived in...