I do them as a supplementary move, with an un-weighted barbell.
Unless your back is in a "touchy" state, it is safe as long as you load it intelligently, just like any other exercise.
When I taught at a former Gymnastic Bodies affliliated gym, we always started students with a pvc pipe threaded through a 5 lb plate. That ought to tell you where to start. If you don't feel anything, just gradually add weight until it feels good.
Some tips I have gathered over time:
Feet:
-keep your weight slightly in your heels
-pay attention to contact points on your feet: more of the inside of the heel and not the outside, the big ball of your foot under your big toe, and the spot just under your pinky toe
-do NOT grip the floor with your toes; this changes the shape of your foot
-your feet don't need to be 100% straight forward, but don't let them turn out
Knees:
-there's no need to lock them out; keep a slight bend in them
Grip:
-depending on what implement you use, you have some options
-I like a barbell, because I have found (at least for me) that a zercher grip does wonders. I highly recommend anyone interested in J-curls try this variation. It opens up the back and encourages scapular upward rotation as you go deeper
-If you don't have a barbell, any other weight will work fine. Just start with something light, as I outlined above.
The movement itself:
-the "goal" is to articulate the spine "one vertebrae at a time." Most people won't be able to do this right away. It takes some time. The tendency is to hinge at first, rather than allow the spine to flex.
-this is why a light load is important; a load that is too heavy will either cause you to tense too much, and thus lose the spinal articulation you are after. Conversely, if you load too much too soon, and relax too much into the move, you can injure yourself.
-let your shoulders relax, so the bar/weight pulls you down
-tuck the chin, then allow the spine to bend down slowly with control under you reach a comfortable depth. Again, think like your spine is a string of beads and you are moving one at a time. Film yourself a few times to see what you look like, and don't forget that it won't look "perfect" at first.
-it is not a fast move: do these fairly slow and with lots of control and intent. Doing them too fast increases the chance you do them incorrectly, or progress your weight too fast.
Set/reps/progression:
-these are up to you. I just do these here and there to maintain, and I don't regularly program it.
-the key with this move is very slow progression over a long period of time. Think of it like gentle mobility, or (very) easy strength, not a max-strength lift. While some people have worked up to bodyweight+ J-curls, they all did so over a period of years (see: Moses Bernard). So while it is possible to get strong with it, it takes a long time and very smart loading and progression. Most critics of the J-curl seem to completely glaze over this notion, citing people for whom it went wrong. It could have gone wrong for many reasons, but my bet is... improper loading and progression, just like everything else!
Hope it helps, happy training.