DataBrowserBox object doesn't seem to have any messages you could use -> you can explore the displaybox structure of the datatable by using Window(dt) << get xml. From that you can get some ideas what you can maybe manipulate to change the width (these can mess some things up, just know that)
Names Default To Here(1);
dt = Open("$SAMPLE_DATA/Big Class.jmp");
wait(1);//demo purposes
Window(dt)[ListBoxBox(1)] << Set Width(300);
There might also be splitter box you could use
-Jarmo