Child support is intended to pay for the costs of raising a child. Typically, these costs include food, shelter, clothing, and health. Life insurance for the guardian could also be an important factor in your child's life if you passed away, and could be considered a good way to invest child support into the child's future.
You should not use your child support to pay for life insurance until and unless the other needs of the child have been met. For example, making sure that there is plenty of food in the home outweighs the importance of having a life insurance policy. Obviously, if you are not able to feed the child today, economic concerns about a possible future are secondary.
Once the basic needs are taken care, life insurance should be considered. If you are in poor health or suffer from a serious recurring illness, life insurance for the child may be an ideal way to make sure that her needs are provided for in the future. That way, even if you cannot be there yourself, your child will still be able to enjoy a healthy, comfortable childhood.
The crux of the matter is that your child's immediate welfare is the first priority, but life insurance is an excellent way to use the money after other needs have been taken care of. Life insurance is not a frivolous expense, after all, and your child, as the named beneficiary, will be the one who receives the value of the policy at some point in the future.
You might even want to set the life insurance policy up to pay into a trust account. This type of account can be set up to pay for housing, care, and even such things as college tuition, doled out in regular payments that are aimed at specific expenses of the child. Trust funds avoid taxation of the policy as it is paid out to your estate as well, allowing your child to receive more of the money, and in a method that gives them the most benefit.
Finally, be aware that term life insurance quotes are typically lower than rates for a whole life insurance policy. Compare quotes and see what pricing different agents can offer,