We use Axis over HTTPS pretty regularly. As Ted mentioned, getting the SSL certificates in order (esp if self-signed) is very important, as it is effectively a deal-breaker.
Code wise, we usually generate our Axis stubs against a plain HTTP service. The call to the ServiceLocator subclass generated by the wsdl2java command will have a "getMyService" method that takes a URL as a parameter in addition to one that takes no parameters. Depending on the exact Axis version and name of your service, the class names may be a little different. But for the most part the following example demonstrates how easy it is to change the URL (HTTPS or HTTP) to point wherever you want upon instantiation of your stub objects.
MyServiceServiceLocator locator = new MyServiceServiceLocator();
MyService_PortType myservice = locator.getMyService(new URL("https://www.myservice.net/MyService.jws"));
Hope that helps.