Hmm, that doesn't make any sense to me.
You were very vague about what this "care" work is. For child care, as in your own children, somebody has to do it. It shouldn't be assumed to be women but it makes no sense for this to be a paid thing.
If you mean something like elder care, for example, and you don't feel it's worth doing it for free, then just don't do it. I don't know why you feel (or you feel anyone else) would be expected to do this work
With a basic income, if you think elder care is important and/or something you want to do, you can afford to do it for free. Because you have a basic income. If you want more than a basic income you have to do something that people want to pay for. But you have the freedom to walk away from any job without fear of homelessness and poverty.
Your worries about permanent slavery seem to have very little to do with basic income. In fact, the main reason for basic income is to free people from the wage slavery we have now. People are stuck in barely livable wages because they don't have the time away from work to get into a better paying career.