Hi guys, we ran into this same problem, upgrading from 2.3.8.5 to 2.3.9.7 and re-exporting Artisteer skins using Artisteer 4.1. The problem is we had styled our search input as Joe Davis explained in his great Styling the Search Input article. However, Artisteer 4.1 forces width:100% on all input[type="text], and that causes the search button to be pushed outside of the SearchInput control.
The fix is to add the following to style-artisteer-overrides (or whichever your css file of choice is, below the main style.css in style.config):
div.art-search input[type="text"] {width:auto;}
Joe, I think perhaps this should be default artisteer override CSS in the exporter, since the mojoPortal search panel is set up with the assumption that both the input box and button are contained within the SearchInput control. Sorry for reporting another potential exporter change just after you pushed out a new version. If you do want to change the exporter and can hold off on publishing a new version until early next week, I will hopefully be done fixing up our sites (just in case I find any other issues). If you think this fix should be handled outside of the exporter, of course I or Joe D. can change the document, so just let me know.
Jamie