Try removing the app from Mac and re-installing it, making sure that you have the latest version of the software. Also be sure that all updates have been installed on your computer. I have Digital Editions version 18.104.22.168765 running properly on my Mac running OS X 10.9.5 and I can get library books properly without crashing. I have had issues in the past with ebook software and removing and re-installing has resolved them both times. Also be sure that you are opening the actual ebook file. When you download an ebook for Digital Editions from the library you will get a .ascm file, that allows you to fulfill the library loan using Digital Editions.
When you download that should go to your downloads folder. You open hat file and it will fulfill you library book loan by opening Digital Editions and downloading the actual ebook. If you are opening the .ascm file it may appear to flash open and then closed because it either a) cannot fulfill the loan for some reason (like the loan being fulfilled once already, even if it didn't work properly), or b) you don't have Digital Editions installed (or it is an old version) so the program cannot open properly once the fulfillment file has been opened.
You apparently only get one shot at a fulfillment, and if there is an issue you have to download the 'book' from the library again, which essentially gives you a new .ascm file to use to fulfill the loan. Once the loan period has expired or the ebook itself has been downloaded by the Digital Editions software the .ascm file is useless and can be deleted.
Which version of OS X are you using on the Mac? Which version of Adobe Digital Editions are you using?
I have Adobe digital editions 4.0 and it comes up with the message "adobe digital editions quit unexpectedly" can anybody help me with this problem. I have a iMac running OS X Yosemite 10.10.1
I found an answer to a similar question on the Forum from TTWTT (TTMTT), who posted a link to download versions 2 and 3 of Digital Editions at Adobe Digital Editions downloads (Adobe Digital Editions downloads). I had tried find these downloads on Adobe's far from helpful website with total lack of success. Version 3 seems to be working on Mavericks; I have no idea if version 3 will work on Yosemite. From other threads in this forum, it sounds like Adobe Digital Editions does not work very well (if at all) on Yosemite.
Whether Adobe is working is a totally different question.
So far it appears that Adobe Digital Editions 4.02 is a disaster, best avoided.