Delta occupies gates in two different terminals (2 & 3) at JFK, so how long it will take you to connect could vary dramatically. You could come in at one gate and leave from one right next to it, for example, in which case it would take you just a minute to get from one to the other. Or you could come in at Gate 8 on Terminal 3 and have to get to Gate 23 on Terminal 2, in which case you'd have to traverse the entire length of both terminals and make the crossing via a walkway between the two in the middle of the process.
One hour is about the minimum I would recommend as a general rule, especially if you have to get an entire family from one gate two another. With little kids that can be even more problematic.
If your initial flight is late and you miss your connection, Delta will work to get you on the next available flight. That means one with enough empty seats for your entire party (unless you are willing to split up, which isn't the case on most family vacations) in the same class of service. That could be in an hour or it could be the next day, you just never know for sure. It's worth checking to see if they have any flights later in the same day to where you are going so that you at least may have options for getting there the same day if you miss your connection. If your connecting flight is the last of the day, you'll know in advance that if you miss it you'll be stuck for the night.
If you did miss your flight because of delays, Delta will just put you on their next flight
