There have been many studies done on the topic with the many finding negative impacts particularly among non-white women.
I'm surprised, given that this has been a fairly well discussed topic, even among liberals, that you are at all familiar.
It shouldn't be surprising that men would find it easier to leave knowing that the government will pick up the financial slack.
The research literature on the effects of welfare on marriage and fertility contains a large number of studies over the last 30 years. The studies use a variety of methodologies, employ several different datasets with different types of individuals, and cover different time periods. Several studies were conducted in the 1970s and early 1980s, but there has been a second wave of studies beginning in the mid-1980s and still under way. Based on the early studies, a consensus among researchers developed a decade or so ago that the welfare system had no effect on these demographic outcomes. However, a majority of the newer studies show that welfare has a significantly negative effect on marriage or a positive effect on fertility rather than none at all. Because of this shift in findings, the current consensus is that the welfare system probably has some effect on these demographic outcomes.