We use a custom theme that adds classes on form elements to the parent div for extra styling control. This generates valid HTML but breaks the jQuery selector used in l10n_client.js '#l10n-client .string-search', because #l10n_client now contains two elements with the sting-search class.
I could not find any good reason why a class selector is used here instead of an ID selector. Changing the selector to '#l10n-client #edit-search' fixes this issue, and is more break-proof, in valid HTML this should never return more than 1 element.
On a side note, changing the name/id of the search input to something a little more namespaced than 'search' might also be a nice idea. It's not outside the realm of posibilities that something else uses 'search' as it's identifier and the Localization client gets loaded on every page, so chances of ending up with 2 elements with id='edit-search' on the same page are fairly real.
I included a patch to change the jQuery selector from '#l10n-client .string-search' to '#l10n-client #edit-search'.