In a perfect world (one with an infinite amount of space to give each orchid a "buffer) it would be ideal to have them not touch each other or splash water between. Not practical... no matter how big one's growing area, it fills up. So it's probably not possible to avoid drips and splashes. I suspect that the odds of cross-contamination are probably fairly small. What caught my eye was mention of "same watering tray" which to me implied that they might be sitting in shared water which would greatly increase the odds of cross contamination. I'd suggest leaving the grates in place, when you water, it drips into a "catch basin" or tray but what drains through one plant isn't picked up by another. Then you can dump the catch basin at the end of the process.
The setup that I used when I did some growing under lights (my spare bedroom was my "greenhouse"
http://orchidcentral.org/GrowingAreas/indoor.jpg , the pans were pretty big, I just watered with a pump sprayer, it ran through the plants, eventually just evaporated (raising humidity). I'd clean out the pans about once in 6 months when they got too icky with algae.