We jump in and swim. When we're done we climb back out. One safety thing I insist on is having a swim ladder that can be deployed from the water, as sometimes you either forget (like when jumping off the bow instead) or might have fallen in (never happened yet). Helps with the climbing out part.
When the crew was having some unreasonable fears similar to yours, I cured her by having a fender tied to a bit of rope floating behind the boat, so she had something to grab onto in case of panic. She got over that pretty quick and now swims far from the boat and doesn't mind how much water is below her feet.
If current is faster than you can swim, don't swim (or wear fins so you can swim faster). You'll surely notice all that water rushing past the boat before getting in.
Definitely do not tie ropes around people, that's bound to go all kinds of wrong. But do let the kids show you a bit more about swimming so you can get over your worries and maybe start to enjoy it. It's fun! Also very helpful in case of problems under the boat.
Only other safety kit I can recommend is a rocket launcher with plenty of ammunition to stop nearby jetskis, powerboats, waterskiers, etc. Those are the real hazards to swimmers, not the water. Otherwise it's swimming only near the boat :-(