Is there a way to delete the key-in browser history/entries?
The key-in history is stored in the User Preference File (upf), so to clear the history, the upf would have to be deleted. For more information, have a look at the following link on the User Preference File
Is this something you want to happen consistantly ? Meaning you never want to see any entries ? If this is the case I don't know a way to do this, as what Conor has suggested will only work when it is deleted. From the point the new one is created, it will start storing it again. Besides, there are other settings stored in the User Preference file as well, and consistantly removing/replacing/deleting this file I don't see as a logical workflow.
why would you want to?
IMHO I think the advantages of it still being there far outweighs any disadvantages that I could think of ...
I would also like to understand what is that exactly you want to achieve as I think, agreeing with Lorys, that having a history of my last key-ins is very handy.
One way of doing this quite quickly, which is not that different from Conor's suggestion but maybe a bit quicker would be to create a second shortcut with the "-restoredefaultsquiet" command line switch and to double click on this whenever you want to delete the key-in history: bear in mind that his will reset Microstation to factory defaults (as basically does deleting the upf) and also clean your temp files.
Here is an article on how to do the above, just do it on a copy of Microstation shortcut called "Reset MicroStation" or something similar.
Restore MicroStation to Factory defaults