What is the ruling on running a café that serves only halal items, where some customers smoke despite the presence of a "No Smoking" sign, and is the owner absolved of responsibility thereby, and is he considered a partner in sin if customers do not lower their gaze from passing women?

There is no harm in opening a coffee shop if it is free from forbidden things, such as smoke and music. The owner of the coffee shop must prohibit smoking in it, in accordance with the hadith: "Whoever among you sees an evil, let him change it with his hand; if he is unable to do so, then with his tongue; if he is unable to do so, then with his heart—and that is the weakest of faith." The owner of the coffee shop should not merely put up a "No Smoking" sign, but should strive to enforce it. If someone sits in the coffee shop and looks at passing women, the sin is on himself, and there is nothing on the owner of the coffee shop.