When one has been using the PICAXE and cable and have successfully downloaded, and then it stops working, it is usually because the last program downloaded is causing the PICAXE to become busy and not see the next download initiation.
That is usually solved by a Hard Reset. When that doesn't work it is most likely the Hard Reset procedure timing is wrong rather than anything else. It can at first be tricky getting the Hard Reset timing right; too fast and the PC will have stopped trying to find the PICAXE by the time it is ready to see it, too slow and the PICAXE will have given up waiting to see if there is a download.
If using PE6 then there is help for getting the timing right. In the Workspace Explorer, usually at the top left, click on the "Compiler" tab at the bottom.
Turn the PICAXE power off. Wait a few seconds. Then click download. When the Workspace Explorer box shows "Connect power now! Searching for hardware on COMx" turn the power on.
That usually fixes things.