Take no offense, but sounds like you're into WYSIWYG editors and have little experience coding and debugging HTML/CSS by hand. Is this true?
For now, assuming you're letting DW manage its own external style sheet sounds like you need to check your HTML script to ensure the style sheet (whatever.css) is linked in the header properly. Could be as simple as a spelling mistake, a path mistake, or missing entirely.
Secondly, not all browsers will render the HTML the same - especially if the layout involves tables instead of DIVs and CSS where content is separated from style and all the code follows DOM standards or isn't XHTML strict. This is a very, very brief explanation as to why, but as to fixing, we must see your code/CSS. If you're not familiar with most of the acronyms I just named, it's time to learn XHTML/CSS and coding by hand. Then you'll be able to use DW for 90% of the work, and be able to debug the expected and usual 10% it and most any other WYSIWYG editor screws up. No editor is perfect because markup is extensible, third party software might be involved, different web technologies, browsers and standards compliancy, and the document type, etc. If such an editor existed which created perfect web sites across all browsers, we'd shut this forum down and all of us would be professional developers using that tool.
-jim