The current market is not free because people are forced to offer their services if they don't want to starve. That's a huge distortion that you would remove with basic income.
At the margin this will certainly cause people to drop out of the workforce. Heck, as I get closer to having enough assets saved to be able to require, I'd probably choose the pull the trigger on that decision sooner.
So we may be removing other large disincentives at the same time.
Later: Which is not intended as more than hand-wavy thinking out loud. Regardless of the impact, a basic income is a political pipe dream in the US. More immediate improvements in quality of life might come from better aligning incentives in the healthcare system and fixing cliffs in existing programs.
This is absolutely true in a free market with no labor distortions (BI, tax, etc). However, we already live in a world with distortions (taxes, welfare, regulations) that drive a wedge between economically optimal and actual choices. How would adding an even bigger wedge help?
Also, due to women's sexual choices, most likely the father is also highly productive. So your "fathers change the diapers" is not really a viable solution - it just means you lose his output rather than hers.