Manage > Additional Settings > Line Weights will allow you to adjust how thick pen #1 (and the others) will be for any given scale. The default values in this table are really just kind of a basic starting point for you to build your desired line weight profile over time by making tweaks to it and incorporating it into your favorite template file. You may deduce from looking at it that just changing a view's scale to a greater ratio may thin up your lines.
I cannot tell if you have anti-aliasing on or off. Big-R > Options > Graphics > Graphics Mode has options relating to this. As you will be able to tell after you look at these options, there is also the possibility that each view will have its own options in the Graphics Display Options dialog box for each view.
In the Print Setup dialog box, try setting the Raster Quality option as high as possible. Also in there, make sure you aren't using the Fit to Page option or a large Zoom value to take a tiny view you've made and print it significantly larger...as that's going to pixilate everything to some degree.
