Let's assume, for the sake of argument, that you have a primary insurance amount of $2,228 (this happens to be what is called the third bend point). In that case, your widow(er) and children would each be eligible to receive up to 75% of your PIA as a monthly payment. However, the total family payment would be limited to about 175% of your PIA, and each survivor's share of that would be prorated. So to continue with the example, if you have one spouse and two children, each would receive 58% of your PIA each month, or about $1300 each.
Your widow(er) would get the payment until the earliest of the following: The youngest kid turns 16, the widow(er) remarries, or they earn more than $48,840 (specific to my hypothetical example). Survivors benefits are subject to the earnings test. In 2019 (future years will be adjusted for inflation) (s)he can earn up to 17,640 with no reduction in benefits. Every two dollars earned above that reduces his or her benefit by one dollar. So in the example, 17640 + 2 * 1300 * 12 = 48840. I believe it's applied annually. But unlike remarriage and kids turning 16, it's not a permanent trigger, meaning if your spouse works one year but not the next, the benefit would be paid in full in the year (s)he didn't work.
Your kids would get the payment until the earliest of: they turn 18, they get married, or they personally earn more than $48,840. The kids benefits are also subject to their own personal earnings test, but it's obviously not common for a minor to earn that much money. As far as I know, the kids' benefits aren't reduced by the widow(er)'s benefits being reduced or eliminated.
$3900 per month is nothing to sneeze at, but considering the level of income you would have had to have such a high PIA, it's probably not enough to keep your family in the manner to which they've become accustomed. So then your widow(er) probably has to work, and then (s)he will not get any benefits due to the earnings test. I've read conflicting info about whether his or her prorated share that (s)he's not even receiving still counts against the family maximum. For example, this article says it doesn't count against the max . The kids' benefits end when they turn 18, just in time for them to head off for college.
There is some kind of adjustment made if you die before you've gotten 35 years of earnings. I honestly don't know how it works, so I can't explain it. But you can log into the SSA website and it will tell you your current PIA but also your specific amounts for survivors benefits.
Your hypothetical family could get around $3300 every month on behalf of the kids, which would definitely be helpful!