Hi Christian,
Unfortunately the screenshots didn’t came through, but I might know what is going on regardless.
There is one known issue with certain IOS and safari versions on displaying the input fields properly. It is sort of a difference on how Safari and other webkit type browsers render some elements.
This custom CSS should resolve it though:
div div.asp_w.ajaxsearchpro input[type="search"].orig {
-webkit-appearance: none !important;
}