Most Zen teachers tell you to keep your eyes open or half-open in zazen.
There are exceptions. Sramana Zhiyi (Chih-i), who greatly influenced the development of Zen Buddhism in China, said that “one should close the eyes” (Essentials of Buddhist Meditation, 85).
Katsuki Sekida says, “I always close my eyes when practicing zazen…. If I want to direct my attention inward, I have to make a deliberate effort to exclude the visual sensations received through the eyes. Closed eyes spare me the difficulty and facilitate inward attention” (Zen Training, 75).
While the research seems to back him up. A paper published in 2020 indicates that “meditation may be most effective with the eyes closed rather than a lowered gaze.”
It is interesting to note that in every sculpture, the Buddha’s eyes are closed or half-closed.
However, for those who are prone to falling asleep during meditation, keeping the eyes open can help. Meditating with your eyes open can also lessen the activity of mental images. In this case, the eyes, half-opened, should be pointing downward at a 45 degree angle.
The bottom line is that there is no sure rule. In the Buddhist tradition you can find advocates of both. So begin with eyes closed. If you get sleepy or have many mental images, open your eyes.
Source: Sprawson, I., Wood, J. & Mantzios, M. “And Now Close Your Eyes or Lower Your Gaze”: Exploring Novice Meditators and Their Attentional Processes During Meditation. J Cogn Enhanc 4, 369–378 (2020). https://doi.org/10.1007/s41465-020-00175-3