I think the main criteria a coach/sensei should look at before allowing his/her students to compete are:
- Can they handle losing? If the kid cries everytime they get thrown in randori, or pinned in ne waza randori, they aren't ready to compete.
- Do they know the rules? Can they follow them?
- Do they have a couple throws they can do in randori?
- Do they have any osae waza they are able to hold with some success? Are they aware of how to escape from the usual ones?
- Break falls of course should be very good, but that's a given i think.
- If the student is an adult, then they should be able to their ego aside and tap when the time is right.
I imagine this list is a little different then ones other people will suggest, but to me, #1 is probably the most important. If a kid doesn't handle losing well, then competing in a tournament will likely be a terrible experience for him/her and their parents. It may even be enough to make them quit.
For a new adult student, there isn't much different from the kids. The ego stuff still stands for sure. As far as physical requirements: if they can't do 5 minutes of randori, they shouldn't be competing. I don't think strength specifically is something that needs to be measured, as you know, judo is about kuzushi, using momentum, and their weight movement. They should be able to last a match without falling over from exhaustion.
It's also important to note that, even if you can do 5 minutes of hard randori no problem, chances are new competitors will find themselves completely gassed after one match, even if they are quite fit. Stress, nerves, and adrenaline all play a big factor.