Generally, to make a pool "safe" you need a minimum 6 foot fence with a "small child unclimable" design - meaning all vertical bars above the bottom crosspiece up to the 6 foot mark with no toeholds to climb, so not chain link or lattice or curlicue design or such - PoolGuard or similar wrought iron, solid cedar plank (board to outside), etc - lots of designs out there - and a padlocked gate.
Other alternatives include filling it in of course, or building a deck across to cover it if you want it usable in the future (though need means to remove deck section to clean out debris and water from time to time), or a safety walkable pool cover which is quite expensive.
I am not keen on the net idea - strangulation and entanglement risk in both the net and the tiedown ropes, plus very hard to tie down so an inquisitive or role-playing pirate or SEAL or such cannot get under the edge. If all kids in the area were say 7+ or so maybe a boarding net concept, firmly anchored all around the edges to the ground so it could be climbed and walked on by kids like in an obstacle course would work, but personally I would not do it with that young a child. If you use a net, you want the black or white nylon or poly-dacron type (feels soft to the touch), not the hard poly type which is hard to fall on and also creates real nasty cuts because it gets sharp ends sticking out as it is exposed to sun. Manila is also good but does not last anywhere as long outdoors.
Unfortunately, I am afraid your cheapest but least desireable solution aesthetically is to fill it in - maybe only temporarily with an easily excavated gravel until the kids reach safe age then back to Koi pond, then back to infilled when grandkids come along, then back to Koi, ...
Most permanent solution keeping the pond would be to treat it as a swimming pool and put a swimming-pool rated fance around it with walk-around/lawn chair room inside, so maybe 16x20 foot or so. Unfortunately, probably in the $3000-6000 range for that solution in wrought iron if professionally installed - maybe in the $1000-2000 range if self-installed for aluminum or wrought iron respectivelyi. Ask a pool installer or fencing company for better estimate. Enclosed cedar board fence or vinyl fence, which would block pond view unless you were inside the fence, probably on the order of $2000-3000 range, ballpark.
Tough choice - and teaching your kid to swim does not solve the problem of accidental fall and being knocked out on the way into the water or of small friends who cannot swim, and of course the whole idea of the fenced back yard is the kids are able to go out and play (or get kicked out to play) at any time without full-time parent overwatch.