Drop Down Lists without the Arrow option can be dropped down using Go To Field to place the cursor into the field. (If this is the first field in your sort order, you don't even need to do that.)
With Pop up Menus or drop downs with the arrow option, that can't be done as far as I know. It'd be a nice feature to have that option in a future version of Filemaker: http://www.filemaker.com/company/feature_request.html
You can use an OnObjectExit script trigger to perform a script that checks to see if the field is still empty when the user clicks or tabs out of the field.