Rising unemployment is partly due to a rise in automation and industrial efficiency. The ultimate limit is, (almost) nobody needs to work, yet goods are still available and continue to increase.
If you use basic income to replace the pension money, but don't confiscate the 401K savings, you've massively disadvantaged the pension holders. If BI is $25K/yr, now one person is living on $75K per year (or $65K/yr and passing along a $1MM inheritance) and the other is living on $50K and will die without a nest egg to pass along.
Trust me, those pensioners are one pretty well-organized voting bloc... ;)
Ideally (state) pensions would become obsolete and retirement funds would become an entirely personal issue (as they've largely become in Germany btw -- the tax-funded pensions will at best barely keep you above the poverty line even if you've worked every day since from 18 to 65).
So as far as pensions go you will have to go with an unfair transitional period. You could freeze the pensions and pay them out in proportion to the (now discontinued) contirbutions (which at least in Germany is how it works already) on top of the BI. But for pensioners you pretty much end up allowing them to "double dip" or drawing some arbitrary cut-off line for who will receive BI or only grant BI in proportion to the pensions.
But as you may have noticed I'm not well-versed when it comes to US pension systems. I'm assuming it's similar to ours but considering even healthcare was purely defined by the employment contracts until fairly recently I wouldn't be surprised if it's drastically different.
Basically, government workers agreed to work for a lower current salary for many years, with the coincident agreement that they'd get X% of that (via some complex formula) for life after Y years of service. That X% provides a modest but certainly comfortable lifestyle.
There's no way you can set basic income levels high enough to provide everyone with that same level of lifestyle before they've lifted a finger of productive work. Basic income must be more basic and spartan than that in order to work mathematically/economically/practically.
(My parents both retired as school teachers with 25-ish years of service and a full pension. I'm not privy to the details, but it's enough for them to live reasonably well, but that didn't fall out of the sky; it came from them raising kids and living 25 years on school teachers' modest salaries.)
I concur that child benefits need not exist (unless/except to the extent that children receive basic income of their own).