You can set up test listeners that run before or after test suite / test case. Please take a look here: https://docs.katalon.com/katalon-studio/docs/test-listeners-test-hooks.html#visualized-workflow.
So you can do something like this:
@BeforeTestCase
def yourBeforeListener(TestCaseContext testCaseContext){
WebUI.openBrowser('your-url.com')
}
@AfterTestCase
def yourAfterListener(TestCaseContext testCaseContext){
WebUI.closeBrowser()
}