It's surprising there can be so many problems with such a simple circuit but it is notable that the overwhelming majority of problems arise when people have built their own circuits or are using non Rev-Ed supplied cables.
I would recommend getting both an official cable, a pre-built prototype board or prototype kit and a 4V5 battery pack which when assembled should be guaranteed to work. An 08M and a protoboard is low cost and can save many hours of frustration and reduces the risk of damaging more expensive PICAXE's. While replacing the PICAXE will be necessary if it has been damaged it may not solve whatever the underlying problem is.
It is easier to replicate a working system than build one from scratch ( though the majority do not seem to have problems doing so ) and a known working system allows comparative tests to be performed. If a home-built cable works as an official cable does then it's almost certain that cable is built right. When everything is home-built there are so many places an error could be ( computer connection, cable wrong, wires crossed, socket miswired, wrong PICAXE pins connected, etc ) it is hard to tell where a problem may actually be. There may be more than one problem to solve and it won't work until all problems are solved.
It's also essential to follow-up on suggestions and report back exactly what the situation is after making changes to help people help yourself. A circuit diagram and photograph can help quickly solve wiring problems ( as was the case with monkeydust ). Repeating the Serial Port Test and reporting voltages read may seem a laborious task but really can help others to identify what is wrong. A detailed "this is what it does currently" is far more useful than, "it didn't work, I changed this, it still doesn't work".
For example, you report the Serial In voltage was 0V when the Serial Port Test on-screen LED was on, 5V when off - the opposite to what would be expected - what voltages are you reading on Serial In now you have swapped the cable wires ?