The short answer is both.
Meditation with the eyes closed helps to bring the focus inward, which is the quickest path the experiencing absolute samadhi or the ground of our awareness without emotional, physical and mental activity. The meditation posture should keep us awake and counting the breath will keep our concentration lightly engaged as we look inward.
Meditation with the eyes open helps us to eventually achieve active samadhi or the experience of the momentary arising and passing away of phenomena in the outer world. It is said that open eyes help to keep the beginning meditator awake throughout meditation and this is true to some extent. However, we can daydream with eyes open just as easily as with eyes closed so paying attention to the breath or some other device for light concentration is crucial in the beginning.
Partial sleepiness during meditation is not a problem. In fact, it is desirable because we learn to be aware of our minds at the threshold of sleep by remaining aware as we approach it. Anyone who has done a long meditation retreat will testify that there is plenty of sleepiness going on in those endless afternoons. Meditating through this sleepiness is what gives us the capacity to be as aware of our minds even when we are tired. It is meditating through this sleepiness which allows us to see that pure consciousness operates independently of the content of our minds.
During these long retreats where many hours are dominated by delta brain waves I find it is beneficial to keep the eyes open because it helps to keep me from dozing off and allows me to see and process what I’m normally unconscious of. There are sometimes very cool moments where we see our dreams (especially ones that we dream repeatedly) arising and passing away while we are awake. It is also cool to see that what seems like a long, in-depth dream when we are asleep occurs in only an instant or two of waking time. Our subconscious is no longer such a mystery.
When we are only doing one or two sits and sleepiness is not an issue, it helps to close our eyes so we can go inward much quicker without distraction. If we have practiced for long enough that waves of discursive thoughts no longer arise, keeping our eyes closed helps us to focus on emotional or physical arisings in our bodies with greater speed and clarity and can help us negate these too and get to “zero” faster.