The maps itself are stored as png. There are usually four map images for a variant, two for each map size (small/large). One of the two images contains the territories of the map where each land territory gets a unique color (for internal purposes), the other contains the names-overlay. You can check this for example for the Classic variant in the files section (which can be reached via the variant page). Things might get a bit more complicated for large maps or FoW-variants.
Maps are already printed at highest possible resolution (each pixel of your screen representing the map contains one pixel of the map image). Therefore I will concentrate on how to improve the "optical appearance".
The software makes it impossible to use anti-aliasing for the internal maps, so a the only way to improve the optical appearance of a map would be to use larger images for the internally stored maps and resize them before they are printed in the browser. Something similar is already achieved for large maps as a "zoom function", but only works on some browsers (e.g. Firefox).
If the goal would be a better optical appearance rather than the "zoom", a better coding solution would have to be made. This might result in more divergence from the original webdip code, but Oli, the site admin, tries to avoid this.
In addition and most importantly, maps would have to redrawn (just rescaling the internal maps will probably not work, since your graphics program will probably introduce some sort of anti-aliasing in this step or you will just get some "enlarged pixels", which wouldn't give any benefit). After the maps are redrawn at higher resolution, the software have to be told to use new coordinates for e.g. drawing units on the board. Both map drawing and inserting territory data are tasks, that can get time-consuming, so making a solution for all the variants would be *very* much work.
Conclusion: A better optical appearance of the maps is difficult if not impossible to be achieved with the current webdip-software. There are some examples, where brilliant map makers achieved very nice-looking maps despite the limitation of no anti-aliasing. But that is not a matter of improving resolution.
If you are interested in variant development itself, the following place is currently the best to start:
http://lab.diplomail.ru