As much as I'm a fan of catching BART to/from SFO (in fact, I'll be doing exactly that later today!), with 4 people your better option is likely to use Uber/Lyft - most likely their larger vehicle services (Uber XL/Lyft XL) given the number of people. This will give the most convenient option, and will likely cost around the same price as other options - although as is always the case with Uber/Lyft that can depend on the time of day as their prices are variable.
The other option is BART as you've suggested. This will likely only be a viable option if your starting point is close to a BART station - otherwise you're still going to need to catch Uber/Lyft/taxi to get to the BART station at which point you might as well catch it all the way to SFO.
As you've stated it is not possible to buy paper BART tickets any more - you will need to have a Clipper Card for each person. For those with suitable mobile phones you can use that - otherwise you will need to buy a clipper card which is available for $3 at all BART stations and from other locations that are listed on the Clipper Card website.
The BART fare from downtown San Francisco to SFO is $10. There are discounts available for children below 18 bringing the fare to $5, however in order to get this discount they will need to have a Youth clipper card, which are only available at very select locations, and require proof of age in order to obtain. It's likely going to be easier to simply get a normal clipper card and pay full fare for the children.
If the adults have suitable mobile phones and the children don't, that brings you to a total of 4x$10 for fares, and 2x$3 for Clipper Cards, or $46. Which is probably similar to what an Uber XL fare is going to be!