This question is asked often, and honestly, there is no definitive current research one way or another. The most that can be said is that, from observation, good poly families are very good for children, and dysfunctional ones are as bad for children as bad monogamous households. This was also the conclusion of the last serious research into the question that we know about (Constantine and Constantine, 1973). There is a need for more research in this area.
Families of all kinds face the challenges and emotional scars that the adults bring to them. A good household of three or more responsible adults can raise kids well more easily than a household of two. Finances are likely to be more stable with more working adults, and full-time home child care becomes more possible, not to mention taxi service to soccer practices and the like.
On the downside, when poly families face financial, job, or relationship stress, it can be amplified because of the multiple people involved and because of mainstream hostility to the family structure.
Another upside: Most successful polyamorous people have great communication skills and the ability to negotiate and work out problems without yelling and throwing things. Such skills are modeled to the children. So, many kids raised in these families are unusually good at communicating their needs and fears to their parents. And spreading parental duties among three, four, or more adults can offer kids not only more support and love, but a bigger collection of parenting skills, than kids often get in the modern (and historically unnatural) nuclear family of two adults only.
Parents in polyamorous families do need to keep the children’s emotional and physical needs at the fore when bringing in any new partners. It is also important that parents be aware of how being in a “different” family affects their kids in school and away from home. As with gay and lesbian parents, kids of polyamorous parents can suffer discrimination and prejudice.
Overall, most children from polyamorous homes seem to be outgoing and well-adjusted.