The above advice is excellent. I have been to Kuwait several times - often alone. I am blonde with blue eyes - very noticeable on the street. Men would stare but few would approach me or say anything. I have been shopping and sightseeing alone and never had any problems. If it is possible, try to stay with a group of people, especially at night. There are women walking alone on the street, but they are few and far between. One gal I work with was stopped by police and sent back to the hotel when she went out for dinner alone. Knowing her, she may have been inappropriately dressed. Definitely wear long sleeves and long pants or a skirt past your knees. Do not wear anything tight or revealing. However, within the confines of the hotel, you can get away with a lot more. I would head to the hotel breakfast in a loose t-shirt and sweatpants and used the hotel pool (although I did wear a one-piece bathing suit and covered up for walking in the public areas of the hotel).
There are a LOT of American military personnel - they were much more likely to approach me. I felt that it wasn't always the best idea to hang out with them or get into extended conversations. Best just to mind your own business. There are definitely mixed feelings among the locals about the military presence.
Overall, I truly enjoyed my trips to KWI and hope to return. Excellent shopping - especially for gold! The hotel staff helped me with shopping and sightseeing and I never felt unsafe. Service was always wonderful and the restaurants were delicious and relatively inexpensive.