you might want to wait till some more guys, from back east chime in and give their .02 about this before you make your decision about this. i personally just buy a new rear axe assm. and do a swap. and before you put the new one in, sand/wire brush it down and primer seal it and then give 2-4 good coats of paint and from then on, was off your underside once a month with fresh water and every summer clean the whole underside good with a wire brush and shoot a good coat of paint on it... you guys like your salt back there and it will eat frames too.
I'm one of those guys back east and have had extensive rust on mine (check the link in my sig). This is what my axle housing looked like before I did the replacement (the clean black parts are where I stripped and painted it before realizing it wasn't salvageable)
As I said, my housing had rusted through in multiple spots on the diff cover area, some significant spots. Here's my take on the safety of it...
The first place that will rust to failure is the diff cover (it's the thinnest part). When this leaks, it will drain your diff of oil, possibly causing a diff failure (probably not a safety issue, but could strand you if it blows). The rest of it I don't think is a big deal for safety, I had thick flakes chipping off everywhere and a good thick layer of corroded, swelled metal underneath that (I could chip it off with a screwdriver if I was aggressive enough). I didn't really see a concern with strength, even on the UCA uprights, which are the next thinnest spot.
I wouldn't be too worried about safety and transporting your family (my pastor has the same issue on his 96 ltd, though not as bad as mine, and I am letting him keep transporting his family in it, or rather, not recommending he doesn't). If you want to keep the vehicle, then it is something that you will want to deal with, and that likely means replacement. My recommendation is to look online (here, yotatech, etc) for someone with a full axle assembly or at salvage yards for a non-rusty one (though you may not be able to find one salvage in the rust belt). At that point it is a pretty easy bolt on job, though you will want a couple helpers if you can get them (it's doable alone, but easier and quicker with help).
You will want to check your LCA's and UCA for rust damage as well, probably replace your shocks at the same time (stockers are only like $35ea for rears), and may want to replace some nuts and bolts, so you'll want to inspect for which ones you will need to replace beforehand and order them.
If you decide to go for it, we can talk more details, and I have some recommendations to make it easier. It really doesn't require alot of skill or knowhow to replace, but it will take some courage, hard work, and a day or two (if all goes well).