The simple way to do this is to use a self-locking screw pitch. Pretty much any single-start thread using a sliding nut cannot be back-driven so the load will not fall. Normal 8x8 trapezoidal thread screws will easily back-drive because of the steep pitch.

Likewise, a worm drive between the motor and Z stage will hold the load. You would want to switch from screws to belts for the main motion stage in that case though, to avoid having too much total gear reduction.

Both of these solutions will limit your maximum Z speed, of course. But they're simple and reliable. Clutches and brakes add a lot of complexity and must be actuated somehow. Designers who want the load to stay suspended almost always simply use single-start screws.

The first, the simplest and the chippest solution (as described by Ryan Carlyle) is to use self locking screw - no doubt.

But I understand you already have your printer. Changing screw (thread rod) is a bit complicated I suppose. As it also needs to change "nuts" and change control settings and so on and so on. So in my opinion, in your case, it's much simpler to change motor to one with break or at least the one with permanent magnets which would hold some torque to support your bed.

Another solution which would also not require to exchange your existing parts is to use kinda ratcher driven by solenoid which would work as a break when deenergized. Where to install it, how to ensure it will hold your bed or prevent your threded rod from rotating is another issue but all options I suggest doesn't need to destroy your existing construction. They all work as additional part.

If you would be so kind and reveal some photos of your design, it would simplify to find proper application of solenoid break.

It should be possible for you to add a counterweight. Pulley at the top, and a string / wire / etc up from the build platform, over the pulley, and back down to a hanging weight. You'll probably want two, one on each side.

This won't directly prevent the motor from slipping - but it will lessen the amount of force trying to make the motor slip.

Note that it will decrease the maximum z-axis acceleration downwards, but will increase the maximum z-axis acceleration upwards. Unfortunately, downwards is the direction that matters for a print.

Also, you could try a circuit to short out the motor when the unit is powered off. The simplest thing would probably be a relay. Note that this could cause issues if the circuit fails in shorted position unless it also disengages the controller input. (This may not be enough to prevent movement, however, just slow it.)

If your stepper motors have shafts protruding from both ends then it may be possible to simply 3d print a fixture that allows a thumb screw to be tightened and clamp down on the unused part of the shaft.

This would however be a very manual approach and would require you to be mindful that the screw must always be disengaged just after motors are powered on..