I was first taught to meditate with my eyes closed. This seems to be common in some Theravadin schools, Vipassana meditation groups and the Triratna Buddhist Community.
I am now meditating in the Tibetan traditions of Kagyu/Nyingma. My main practice is Formless Meditation (meditation without an object of focus). In this meditation, the instruction is to meditate with the eyes open, or half open with the vision cast down a little. The gaze is kept soft and you do not focus on any particular object . You can experience this by trying to see through your peripheral vision.
So why do some Buddhist meditation schools recommend eyes closed and some eyes open?
When I learnt to meditate with the Triratna Buddhist Community I was told that the visual input was disturbing, particularly to beginners and that shutting my eyes would help me to not get distracted aid access to super-normal states of concentration, called Dhyana. This seemed reasonable and I practised this way (and taught meditation this way) for many years.
Later, when I was introduced to Formless Meditation (within the view of Dzogchen/Mahamudra) I was taught that meditating with eyes closed had the side effect of turning your meditation into some kind of inner-world that actually cuts you off from a wider, more inclusive reality. It can become a comfy place that you go to for refuge from a busy and disturbing life, but it is not a place that you can remain. The benefits of this being quickly lost when you return to the wider world.
The advantage here of meditating with eyes open is that you remain connected and open to the world. Therefore, you do not require any transition into daily life. with practice you can meditate in situations that may appear to be counter productive, such as at the airport or on a busy train. I once heard Ato Rinpoche say that he liked to test his meditation in noisy, stimulating airport lounges.
Also, with Formless Meditation, part of the instruction is to soften and open out into the space around you. Meditating with eyes closed gives an auditory and felt sense of space, but meditating eyes open adds another dimension to this. You really need to experiment with this for yourself and see what differences there are for you.
Formless Meditation also emphasises a balance between Shamatha (concentration) and Vipassana (insight) right from the get go. The view being that the Truth of things is closer than we might think and that simply resting the mind without contrivance on our present experience is sufficient to lead to insight, without the need for deep Dhyanic experiences. Indeed, Chogyam Trungpa Rinpoche warned students about becoming stuck in the attractions of "Shamatha's pool". The experience of Dhyanic concentration can be very pleasurable leading meditation students stuck in trying to replicate previous meditation experiences.
I think the view of the Mahayana/Vajrayana of practising Buddhism and meditation for the sake of all beings (the Bodhisattva path) may tend towards an eyes open meditation technique simply because it is orientated outwards into the world. By not separating our meditation into an internal matter, we stay connected to others and can more easily carry the benefits into our daily interactions.
So who is right? Meditating with eyes open or eyes closed? The only way to know is to try both methods for yourself. This can be done in one session of meditation - perhaps starting eyes closed while tuning into the body and then opening the eyes for the main practice. Alternatively, trying each method for a period of some weeks or months, perhaps keeping a meditation diary to note and remember the effects.
Whichever method, the best results will be obtained by exploring your experience with a qualified meditation teacher or mentor.
In this way we can talk about our meditation practice open up areas for closer examination in a way that we could not do on our own. This will save us valuable time lost down dead-ends and help us to maintain a lively, effective and authentic meditation practise.