DWR applications are normal web applications, and so they can be tested and debugged using normal debug tools available in the Eclipse IDE. The Web has information about debugging in Eclipse, for example, the article on the IBM developerWorks site (http://www.ibm.com/developerworks/opensource/library/os-ecbug/). Several books about Eclipse and its features are also available.
Another method for debugging is logging. It may sound "old-fashioned", but here is a quote from the book The Practice of Programming by Brian W. Kernighan and Rob Pike (this quote is found also from Apache's Log4J website):
As personal choice, we tend not to use debuggers beyond getting a stack trace or the value of a variable or two. One reason is that it is easy to get lost in details of complicated data structures and control flow; we find stepping through a program less productive than thinking harder and adding output statements and self-checking code at critical places. Clicking over statements...