How do I get the dropdowns in Archives and Catagories widgets to work?

Did you check the Display as dropdown checkbox?

Yes. And there is a selection list showing in the widgets. They just don’t work.

I’ve checked your site and I can’t find any JavaScript errors, but I can’t reproduce it on my test server. Try disabling your plugins one by one and see if it changes anything.

Yes–you’re right. It was a plugin that was preventing selection. It is fixed now.

I also added styling for the postform selection lists in my child style.css:

/* ~~~ SELECTION LISTS IN WIDGETS  ~~~~~~~~~~~~ */

select,
input[type="postform"],
textarea {
	color: #666;
	border: 1px solid #ccc;
	border-radius: 3px;
}

select:focus,
input[type="postform"]:focus,
textarea:focus {
	color: #111;
	box-shadow: 0 0 5px rgba(162, 196, 251, 1);
	border: 1px solid rgba(162, 196, 251, 1);
}

select,
input[type="postform"] {
	padding: 3px;
}