How do I increase the resize margin on windows?

As others have mentioned, it depends on the theme. Try a few out. What I frequently do instead is hold down Alt and middle-click (right-click in some environments) and drag to resize, which works anywhere on the window.

Note that many themes that seem to have a 1px border (or no border at all) actually do tend to have a generous invisible border just outside the window. You usually don't have to pinpoint a tiny window border to resize.

To modify Ambiance to have a wider margin, open /usr/share/themes/Ambiance/metacity-1/metacity-theme-1.xml and increase the values of the following properties:

<distance name="left_width" value="1"/>
<distance name="right_width" value="1"/>
<distance name="bottom_height" value="1"/>

You may want to back up the original file before modifying, and/or make a duplicate of the entire theme.

Note that you can also resize windows by pressing Alt + Button2.

There is also a default keyboard short-cut in gnome -- Alt-F8 -- that is a 'resize window' function.