Are you getting the .acsm files downloaded by the browser, but then not opened by ADE?
Try starting ADE, then drag-dropping the .acsm files from the downloads directory in your file manager (Windows Explorer, Mac Finder) to the running ADE.
If that works and you are on Windows, it may be the file association is broken.
To make it work more easily in future ...
Sometimes the registration of ADE to Windows as the default hander for .acsm and .epub files does not work;
that is what causes the double-click in Explorer to fail; and usually prevents opening from the browser as well.
especially in Windows 8/8.1, and maybe 7;
this appears to be a bug in Windows (for once not a bug in ADE).
Two possible ways round it.
First, try to force it in Explorer
Right-click on a .epub file (and repeat entire sequence for .acsm if necessary)
Select 'open with' if on the list, 'open' if not.
Select 'choose default program'
Select ADE if it is on the list,
If not select 'more options' and navigate to the ADE executable (may be at C:\Program Files\Adobe\Adobe Digital Editions\digitaleditions.exe)
Now .epub/.acsm files should be associated with ADE, and double-clicking should work.
Sometimes, Windows allows you to go through all that, but then refuses to make the association you have asked for.
In that case, the only way I know is to use a third party file association program.