This topic has been discussed over and over again, hence you might find a reluctance for some members to contribute to this post. I don't mind giving you a starter 10.

Probably the best way to get your guys involved is during site inspections, walkabouts, etc. Show them an unsafe condition which is the majority of near miss reporting and explain how a simple report can prevent someone from harm. You could also explain that unsafe acts can be a near miss by putting people at danger - although workers are reluctant to 'grass' on other colleagues.

Some companies have offered incentives for workers to raise near misses, others have introduced quotas. Don't like quotas because you get loads of silly and repeat things which are not worth the trouble as you do with incentives to a lesser degree. I would start by making sure your supervisors and managers are on board and lead by example.

Finally, people like to see something positive come of raising near misses in the form of a reply and individual feedback of the action which was taken to close out the near miss. A notice board in a prominent position for those which are anonymous. Make sure near miss forms are readily available and also different mediums can be used i.e. email, texts, etc.